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✓ 7 results · all verified · 0 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

MCMC central limit theorems, MCSE, and long-run variance

1 · Prerequisites

2 · Summary

Summary

This page defines covariance long-run variance, three stationary mixing coefficients, geometric and polynomial total-variation rates, one-step minorization, and the overlapping-batch-means estimator. It proves the kernel conditional-moment lemma, rate-to-mixing bounds, measurable-factor bounds, the degenerate zero-variance reduction, and the Poisson-equation martingale CLT with its covariance identity.

The six MCMC CLT regimes from Jones (2004) are proved locally with their moment, reversibility and rate-envelope hypotheses. Their suppliers prove the bounded and unbounded strong-mixing limits, the rho and uniform-mixing limits, the reversible geometric correlation bound and the integrable geometric envelope. The Flegal–Jones OBM contract retains every numerical batch-size condition and corrects the source's incompatible simultaneous monotonicity requirements for integer batches. Local Gaussian quadratic-form bounds and a pathwise Brownian-approximation comparison prove the OBM reduction, including zero approximation variance and the exact floor-power batch calculation. The full Markov-chain OBM theorem combines the positive-variance strong approximation with the separate zero-variance strong rate, applies under arbitrary initial laws, and identifies its covariance long-run variance through the stationary mixing CLT. A finite second moment alone is not presented as a CLT criterion, and covariance summability is not used to identify a CLT variance without a theorem.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Flegal–Jones overlapping-batch size conditions

Definition

Let $δ > 0$ and let $(b_{n})_{n \geq 2}$ be positive integers. Say that the sequence satisfies the Flegal–Jones OBM batch-size conditions for exponent $δ$ , with the integer monotonicity correction described below, when $b_{n} < n$ for all sufficiently large $n$ , $b_{n} \to \infty$ , $n / b_{n} \to \infty$ , and there is a constant $c \geq 1$ such that $\sum_{n = 2}^{\infty} (b_{n} / n)^{c} < \infty,$ while $\frac{b_{n}}{n} \log n \to 0, \frac{n^{2 / (2 + δ)} (\log n)^{3}}{b_{n}} \to 0,$ there are $n_{0} \geq 2$ and $c_{1} < \infty$ with $\log n / b_{n} \leq c_{1}$ for all $n \geq n_{0}$ , and $b_{n}^{2} n^{- 2} \log \log n \to 0, b_{n}^{4} n^{- 3} \log \log n \to 0.$

These are all the numerical rate, summability and endpoint hypotheses of Flegal–Jones Theorem 2. The source additionally asks that both $b_{n}$ and $n / b_{n}$ be nondecreasing. Those two requirements are omitted here: for an integer sequence with $b_{n} < n$ eventually they are incompatible with $b_{n} \to \infty$ . Indeed, ratio monotonicity would imply $n (b_{n + 1} - b_{n}) \leq b_{n} < n$ , while integer monotonicity would force the nonnegative integer $b_{n + 1} - b_{n}$ to be zero. Thus the literal source conditions would make $b_{n}$ eventually constant. The corrected interface preserves the intended floor-power batch sequences, and the Brownian quadratic-form proof of OBM consistency needs neither monotonicity assumption.

This is a sufficient-condition interface; failure of a condition does not imply inconsistency of the estimator.

For the floor-power sequence $b_{n} = ⌊ n^{ν} ⌋$ , the intended source range $δ > 2 / 3$ and $2 / (2 + δ) < ν < 3 / 4$ satisfies this corrected contract. Indeed, eventually $n^{ν} / 2 \leq b_{n} \leq n^{ν}$ . Choose $c > \max {1, 1 / (1 - ν)}$ to bound the summability series by $\sum_{n} n^{- c (1 - ν)} < \infty$ . The other displayed quantities are bounded respectively by $n^{ν - 1} \log n$ , $2 n^{2 / (2 + δ) - ν} (\log n)^{3}$ , $2 n^{- ν} \log n$ , $n^{2 ν - 2} \log \log n$ and $n^{4 ν - 3} \log \log n$ , all tending to zero. Also $b_{n} \to \infty$ , $n / b_{n} \to \infty$ and $b_{n} < n$ eventually. These checks establish the numerical contract without asserting the impossible simultaneous monotonicity.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Geometric and uniform ergodicity for MCMC CLTs

Definition

Let $P$ be a probability kernel with invariant probability $π$ on a measurable space $(X, X)$ . Write $P^{n} (x, \cdot)$ for its $n$ -step transition probability and use $∥ μ - ν ∥_{T V} = \sup_{A \in X} ∣ μ (A) - ν (A) ∣$ for probability measures.

The chain is geometrically ergodic if there are a measurable finite function $M : X \to [0, \infty)$ and $r \in (0, 1)$ such that $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) r^{n} (x \in X, n \geq 1) .$

It is uniformly ergodic if this bound holds with $M$ bounded on $X$ . These are convergence-rate properties, not conclusions about a particular function's CLT or long-run variance. If a proof needs $\int M d π < \infty$ , that integrability is an additional stated hypothesis unless a cited theorem establishes it.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Long-run variance for a stationary Markov chain

Definition

Let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with invariant probability $π$ , and let $f$ be real Borel with $π (f) = 0$ and $π (f^{2}) < \infty$ . For $k \geq 0$ , set $γ_{k} : = {Cov}_{π} (f (X_{0}), f (X_{k}))$ . When the Markov-chain CLT $n^{- 1 / 2} \sum_{k = 0}^{n - 1} f (X_{k}) \Rightarrow N (0, σ_{f}^{2})$ holds with finite variance parameter and $\sum_{k \geq 1} ∣ γ_{k} ∣ < \infty$ , the covariance long-run variance is $σ_{L R}^{2} (f) : = γ_{0} + 2 \sum_{k = 1}^{\infty} γ_{k} .$

This definition does not infer a CLT from covariance summability, or infer that a weak-limit variance equals the covariance sum. Whenever $σ_{L R}^{2} (f)$ is used as the variance parameter in a CLT or MCSE claim, that equality must be established by the theorem being applied. For example, the local Poisson-equation theorem proves the equality under its square-integrable solution hypotheses.

Remarks

Under stationarity, the finite variance identity is $\frac{1}{n} {Var}_{π} ⁣ (\sum_{k = 0}^{n - 1} f (X_{k})) = γ_{0} + 2 \sum_{k = 1}^{n - 1} (1 - \frac{k}{n}) γ_{k} .$ Absolute summability makes the right side converge to $σ_{L R}^{2} (f)$ by dominated convergence. Equality between this variance limit and a variance parameter from weak convergence is a separate assertion; it is not part of this definition.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Strong, uniform, and maximal-correlation mixing coefficients

Definition

Let $Y = (Y_{k})_{k \geq 0}$ be a process on a probability space whose finite-dimensional laws are invariant under common nonnegative time shifts; call this strict stationarity. For $j \geq 0$ , set $F_{j}^{-} : = σ (Y_{0}, \dots, Y_{j})$ and $F_{j}^{+} : = σ (Y_{j}, Y_{j + 1}, \dots)$ . For $n \geq 1$ , define the strong-mixing coefficient by $α_{Y} (n) : = \sup_{j \geq 0} \sup_{\begin{array}{c} A \in F_{j}^{-} \\ B \in F_{j + n}^{+} \end{array}} ∣ P (A \cap B) - P (A) P (B) ∣ .$

Define the uniform-mixing coefficient by $ϕ_{Y} (n) : = \sup_{j \geq 0} \sup_{\begin{array}{c} A \in F_{j}^{-} \\ P (A) > 0 \end{array}} \sup_{B \in F_{j + n}^{+}} ∣ P (B ∣ A) - P (B) ∣ .$

For square-integrable real random variables $U, V$ with positive variances, write $Corr (U, V) : = \frac{Cov (U, V)}{\sqrt{Var (U) Var (V)}} .$ For each $n \geq 1$ , let $C_{n}$ consist of the triples $(j, U, V)$ such that $j \geq 0$ , $U, V$ are real and square-integrable, $U$ and $V$ are measurable with respect to $F_{j}^{-}$ and $F_{j + n}^{+}$ respectively, and both have positive variance. Define the maximal-correlation coefficient by $ρ_{Y} (n) : = \sup ({0} \cup {∣ Corr (U, V) ∣ : (j, U, V) \in C_{n}}),$ where square-integrability is in the sense of The space $L^{p} (μ)$ as the quotient by null functions. Thus $ρ_{Y} (n) = 0$ when there is no admissible pair, as for a constant process. The process is called strongly mixing, uniformly mixing, or rho-mixing, respectively, when the corresponding coefficient tends to zero. These coefficients are distinct conditions; no implication between them is included in this definition.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Polynomial ergodicity with an integrable rate envelope

Definition

Let $P$ be a probability kernel with invariant probability $π$ . For $m > 0$ , say that $P$ has polynomial total-variation rate of order $m$ with an integrable envelope if there is a measurable $M : X \to [0, \infty)$ with $π (M) < \infty$ such that $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) n^{- m} (x \in X, n \geq 1) .$

This is the specific envelope condition used with the polynomial-rate CLT cases on this page. It does not claim that every definition of polynomial ergodicity in the literature includes integrability of $M$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

State-dependent one-step minorization

Definition

Let $P$ be a probability kernel on $(X, X)$ . A state-dependent one-step minorization is supplied measurable data $s : X \to [0, 1]$ and a probability measure $Q$ on $(X, X)$ such that $P (x, A) \geq s (x) Q (A) for every x \in X, A \in X .$

The function $s$ need not have a positive uniform lower bound; this is the exact one-step assumption used in the overlapping-batch-means theorem below.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Kernel conditional expectations for square-integrable Markov observables

Statement

Assume AC. Let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with probability kernel $P$ , invariant law $π$ , and natural filtration $F_{k} = σ (X_{0}, \dots, X_{k})$ . Let $g \in L^{2} (π)$ (The space $L^{p} (μ)$ as the quotient by null functions) and fix a finite real measurable representative, also denoted by $g$ . Define $P g (x) = \int g (y) P (x, d y)$ and $P (g^{2}) (x) = \int g (y)^{2} P (x, d y)$ on the $π$ -full measurable set where the respective integrals are finite, and set these functions to zero on the exceptional set. Then, for each $k \geq 0$ , $E [g (X_{k + 1}) ∣ F_{k}] = P g (X_{k}) .$ $E [g (X_{k + 1})^{2} ∣ F_{k}] = P (g^{2}) (X_{k}) a.s.$ Consequently, if $D_{k + 1} = g (X_{k + 1}) - P g (X_{k})$ , then $(D_{k + 1})$ is a square-integrable martingale-difference sequence and $E [D_{k + 1}^{2} ∣ F_{k}] = P (g^{2}) (X_{k}) - (P g (X_{k}))^{2} a.s.$

Facts & Assumptions

Given: AC, a stationary chain $X$ with kernel $P$ and invariant law $π$ , its natural filtration, and a real measurable representative $g$ with $π (g^{2}) < \infty$ .

[A1]

AC is the choice principle used by the library's conditional-expectation class convention. (The Axiom of Choice)

[F1]

For a bounded measurable $h$ and a $P$ -chain, $E [h (X_{m + n}) ∣ F_{m}] = P^{n} h (X_{m})$ ; in particular, at $n = 1$ the conditional mean is the kernel integral. (Chapman-Kolmogorov equations)

[F2]

Increasing nonnegative inputs have increasing conditional-expectation versions converging almost surely to the conditional expectation of their limit. (Conditional monotone convergence)

[F3]

Invariance is the event identity $\int P (x, A) π (d x) = π (A)$ for every measurable $A$ . (Invariant and stationary distribution for a Markov kernel)

[F4]

Nonnegative increasing functions pass through an integral by monotone convergence. (Monotone convergence for the integral)

[F5]

Kernel integration of a nonnegative product-measurable function is measurable, with infinity permitted. (Measurability of integration against a kernel)

[F6]

Conditional expectation is linear and preserves order. (Basic algebra and order properties of conditional expectation)

[F7]

Conditional Jensen gives $∣ E [Z ∣ G] ∣^{2} \leq E [∣ Z ∣^{2} ∣ G]$ for square-integrable $Z$ . (Conditional jensen inequality)

[F8]

On a probability space, Hölder gives $π (∣ g ∣) \leq π (g^{2})^{1 / 2} < \infty$ . (Holder's inequality for integrals, including the endpoint cases)

[F9]

An integrable $F_{k}$ -measurable factor can be taken out of a conditional expectation when the products are integrable. (Taking out what is known)

[F10]

Products of square-integrable random variables are integrable by Hölder's inequality. (Holder's inequality for integrals, including the endpoint cases)

Proof

1.1A1F1F5given

For bounded measurable $h$ , [F1] with $n = 1$ gives $E [h (X_{k + 1}) ∣ F_{k}] = P h (X_{k})$ . Kernel measurability [F5] makes the right side measurable.

2.1F2F3F4F6F8step 1.1

Write $g = g^{+} - g^{-}$ . By [F8], both parts are integrable. For $N \geq 1$ let $g_{N}^{\pm} = g^{\pm} \land N$ . These functions are bounded and increase to $g^{\pm}$ . Step 1.1 gives the conditional kernel identity for each $g_{N}^{\pm}$ ; [F2] makes the conditional expectations increase to $E [g^{\pm} (X_{k + 1}) ∣ F_{k}]$ , while [F4] makes $P g_{N}^{\pm} (X_{k}) ↑ P g^{\pm} (X_{k})$ . Invariance [F3], first for nonnegative simple functions and then by [F4], gives $E [P g^{\pm} (X_{k})] = π (g^{\pm}) < \infty$ ; hence both limits are finite almost surely. Subtracting the two finite identities using [F6] proves $E [g (X_{k + 1}) ∣ F_{k}] = P g (X_{k})$ .

2.2F2F3F4step 1.1

Apply the same argument to the nonnegative integrable function $g^{2}$ : the truncations $g^{2} \land N$ increase to $g^{2}$ , and invariance gives $E [P (g^{2}) (X_{k})] = π (g^{2}) < \infty$ . Thus $E [g (X_{k + 1})^{2} ∣ F_{k}] = P (g^{2}) (X_{k})$ . These identities do not depend on the chosen representative: if two representatives differ on a measurable $π$ -null set $A$ , invariance gives $π (P 1_{A}) = 0$ , hence $P (x, A) = 0$ for $π$ -almost every $x$ ; their kernel integrals agree there, and their evaluations at each $X_{k}$ agree almost surely.

3.1F3F6F7F9F10step 2.1step 2.2∎

By [F7] and step 2.2, $(P g (X_{k}))^{2} \leq P (g^{2}) (X_{k})$ almost surely, so stationarity and [F3] imply $E [(P g (X_{k}))^{2}] \leq π (g^{2})$ ; therefore $D_{k + 1} \in L^{2}$ . Step 2.1, the preceding square-integrability bound, conditional linearity and [F9] for the $F_{k}$ -measurable factor $P g (X_{k})$ give $E [D_{k + 1} ∣ F_{k}] = 0$ . The product $g (X_{k + 1}) P g (X_{k})$ is integrable by the two $L^{2}$ bounds and [F10], so [F9] also gives its conditional expectation as $(P g (X_{k}))^{2}$ . Expanding the square and using step 2.2 now yields $E [D_{k + 1}^{2} ∣ F_{k}] = P (g^{2}) (X_{k}) - (P g (X_{k}))^{2}$ . Thus the sequence is a square-integrable martingale-difference sequence.

Source notes

Jones §2 defines the kernel operator $P g$ and states the conditional one-step kernel convention. The bounded-to-square-integrable extension and the exceptional-set version are proved here; no pointwise identity is asserted outside a $π$ -full set.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

A kernel has a countably generated stable sub-sigma-field containing a given observable

Statement

Let $P$ be a probability kernel on an arbitrary measurable space $(X, F)$ , and let $f : X \to R$ be measurable. There is a countably generated sub-sigma-field $G \subseteq F$ such that $f$ is $G$ -measurable and $P$ , restricted to events in $G$ , is a probability kernel $P_{G}$ on $(X, G)$ . For every $n \geq 0$ , $x \in X$ , and $B \in G$ , $P_{G}^{n} (x, B) = P^{n} (x, B) .$ If $π$ is invariant for $P$ , then $π ∣_{G}$ is invariant for $P_{G}$ .

Facts & Assumptions

Given: The measurable space, probability kernel, and real measurable function in the Statement.

[F1]

Measure kernel and probability kernel: $P (x, \cdot)$ is a probability and $x \mapsto P (x, B)$ is measurable for every $B \in F$ .

[F2]

Dynkin's pi-lambda theorem: a lambda-system containing a pi-system contains the sigma-field it generates.

[F3]

Monotone convergence for the integral: increasing nonnegative integrands have increasing integrals with the expected limit.

[F4]

Iterated transition kernels: iterated transitions are defined by kernel integration, with the identity kernel at time zero.

[F5]

Invariant and stationary distribution for a Markov kernel: invariance means $\int P (x, B) π (d x) = π (B)$ for every measurable $B$ .

[F6]

A measurable function between measurable spaces: a real function is measurable when inverse images of Borel sets are measurable.

[F7]

Every nonnegative measurable function admits an explicit increasing sequence of simple approximations: nonnegative measurable functions have increasing nonnegative simple approximations.

Proof

Given: The hypotheses in the Statement.

1.1F1F6constructgiven

Let $A_{0}$ be the algebra generated by ${{f < q} : q \in Q}$ . Inductively, let $A_{j + 1}$ be the algebra generated by $A_{j}$ and all sets ${x : P (x, A) > q}$ with $A \in A_{j}$ and $q \in Q$ . These are subsets in $F$ by [F1] and [F6]. Each algebra is countable: enumerate its generators and then all finite Boolean expressions in them. This also explicitly enumerates $A = ⋃_{j \geq 0} A_{j}$ . Put $G = σ (A)$ . Since rational half-lines generate the real Borel sigma-field, $f$ is $G$ -measurable.

2.1F1F2step 1.1

For $A \in A$ , all rational superlevel sets of $P (\cdot, A)$ belong to $G$ by construction. The class $D$ of $B \in G$ for which $P (\cdot, B)$ is $G$ -measurable contains $A$ and $X$ . It is closed under complements because $P (x, B^{c}) = 1 - P (x, B)$ and under countable disjoint unions because $P (x, ⋃_{l} B_{l}) = \sum_{l} P (x, B_{l})$ is a measurable increasing limit. Thus $D$ is a lambda-system, and [F2] gives $D = G$ . The restriction $P_{G}$ therefore is a probability kernel.

3.1F3F4F7step 2.1

For every nonnegative $G$ -measurable $g$ , integration against $P_{G} (x, \cdot)$ agrees with integration against $P (x, \cdot)$ : this holds for indicators by restriction, for nonnegative simple functions by finite additivity of the integral, and for general $g$ by [F7] and [F3]. Both integrals are allowed to be infinite. Induction in [F4], using this integration identity and step 2.1, gives $P_{G}^{n} (x, B) = P^{n} (x, B)$ for every $B \in G$ , including $n = 0$ .

4.1F5step 3.1step 1.1∎

If $π$ is invariant, the same restriction identity for integrals gives $\int P_{G} (x, B) (π ∣_{G}) (d x) = \int P (x, B) π (d x) = π (B)$ for $B \in G$ . This is invariance of $π ∣_{G}$ by [F5]. All generating enumerations in step 1.1 were explicit, so this construction introduces no choice assumption.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Centered Gaussian quadratic forms have dimension-free even-moment bounds

Statement

Assume AC. For every positive integer $r$ there is a finite constant $C_{r}$ , depending only on $r$ , such that for every integer $n \geq 1$ , independent standard-normal variables $U_{1}, \dots, U_{n}$ , and real symmetric $n \times n$ matrix $A$ , $E {∣ U^{T} A U - tr A ∣}^{2 r} \leq C_{r} {(\sum_{i, k = 1}^{n} A_{i k}^{2})}^{r} .$ In particular $E (U^{T} A U) = tr A$ . The matrix may be singular or indefinite.

Facts & Assumptions

Given: AC, a positive integer $r$ , independent standard-normal coordinates of $U \in R^{n}$ , and a real symmetric matrix $A$ with $n \geq 1$ .

[A1]

AC is assumed as in The Axiom of Choice and the normal interfaces below.

[F1]

A self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis. Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

[F2]

Orthogonal transformations preserve independence and the standard-normal coordinate laws. Orthogonal transformations preserve the standard normal product law

[F3]

The standard-normal law has density $e^{- z^{2} / 2} / \sqrt{2 π}$ . Standard normal and normal laws

[F4]

Expectations of integrable products of functions of independent variables factor. Expectations factor over finite products of independent random variables

Proof

1.1A1F1F2F3givenalgebra

By [F1] choose an orthogonal matrix $O$ with $A = O diag (λ_{1}, \dots, λ_{n}) O^{T}$ . By [F2], $Z = O^{T} U$ has independent standard-normal coordinates. Integration by parts in the density [F3], with $z e^{- z^{2} / 2} \to 0$ at both infinities, gives $E Z_{i}^{2} = 1$ . Every absolute moment of $V_{i} = Z_{i}^{2} - 1$ is finite: for an integer $k \geq 1$ , $∣ z^{2} - 1 ∣^{k} \leq (1 + z^{2})^{k}$ , and this polynomial is integrable against $e^{- z^{2} / 2}$ . Thus $E V_{i} = 0$ , $E (U^{T} A U) = \sum_{i} λ_{i} = tr A$ , and $U^{T} A U - tr A = \sum_{i} λ_{i} V_{i}$ .

2.1F4step 1.1algebra

Expand the even power $(\sum_{i} λ_{i} V_{i})^{2 r}$ over ordered tuples of $2 r$ indices. By [F4], a term whose tuple has an index appearing exactly once has expectation zero. Group all other terms by the partition of their $2 r$ positions into equal-index blocks. Such a partition has block sizes $k_{1}, \dots, k_{q} \geq 2$ with $\sum_{a} k_{a} = 2 r$ . For each fixed partition, taking absolute values and dropping the restriction that different blocks have distinct indices bounds its contribution by $\prod_{a = 1}^{q} E ∣ V_{1} ∣^{k_{a}} \prod_{a = 1}^{q} \sum_{i} ∣ λ_{i} ∣^{k_{a}}$ . All these quantities are finite by step 1.1.

3.1step 1.1step 2.1algebra

Put $L = (\sum_{i} λ_{i}^{2})^{1 / 2}$ . If $L = 0$ , the centered form is identically zero and the bound holds. Otherwise $∣ λ_{i} ∣ / L \leq 1$ and $\sum_{i} (∣ λ_{i} ∣ / L)^{2} = 1$ , so for every integer $k \geq 2$ , $\sum_{i} ∣ λ_{i} ∣^{k} \leq L^{k}$ . Consequently each partition contribution in step 2.1 is bounded by $L^{2 r} \prod_{a} E ∣ V_{1} ∣^{k_{a}}$ . There are finitely many partitions of the fixed set of $2 r$ positions; the sum of these moment products over the partitions without singleton blocks is a finite constant $C_{r}$ depending only on $r$ .

4.1step 1.1step 3.1algebra∎

Since $2 r$ is even, the expanded power is the absolute $2 r$ th power. Summing the bounds of step 3.1 therefore gives $E ∣ U^{T} A U - tr A ∣^{2 r} \leq C_{r} L^{2 r}$ . Finally $L^{2} = tr (A^{2}) = \sum_{i, k} A_{i k}^{2}$ by symmetry and orthogonal diagonalization. This proves the bound and the expectation assertion, including $A = 0$ , singular matrices and matrices of either sign.

TheoremStatement: AI-adaptedProof: AI-generatedjudge pass (gpt-6-luna)Open item page →

Summable conditional projections give a stationary martingale CLT

Statement

Assume AC. Let $T$ be an invertible bimeasurable probability-preserving transformation on $(Ω, A, Q)$ such that $Q (A △ T^{- 1} A) = 0$ implies $Q (A) \in {0, 1}$ . Let $X_{0} \in L^{2} (Q)$ be real and centered, put $X_{k} = X_{0} \circ T^{k}$ for $k \in Z$ , and put $F_{k} = σ (X_{i} : i \leq k)$ . Suppose $\sum_{j = 1}^{\infty} ∥ E [X_{j} ∣ F_{0}] ∥_{2} < \infty .$ Then the $L^{2}$ series $H_{0} = \sum_{j \geq 1} E [X_{j} ∣ F_{0}]$ converges. Put $H_{k} = H_{0} \circ T^{k}$ and $D_{k} = X_{k} + H_{k} - H_{k - 1}$ . The $D_{k}$ are stationary square-integrable martingale differences for $F_{k}$ , and for $n \geq 1$ , $S_{n} : = \sum_{k = 1}^{n} X_{k} = \sum_{k = 1}^{n} D_{k} + H_{0} - H_{n}, ∥ H_{0} - H_{n} ∥_{2} \leq 2 ∥ H_{0} ∥_{2} .$ With $τ^{2} = E [D_{1}^{2}]$ , $S_{n} / \sqrt{n} \Rightarrow N (0, τ^{2}),$ including $τ^{2} = 0$ . If additionally $\sum_{j \geq 1} ∣ Cov (X_{0}, X_{j}) ∣ < \infty$ , then $τ^{2} = E [X_{0}^{2}] + 2 \sum_{j = 1}^{\infty} Cov (X_{0}, X_{j}) .$

Facts & Assumptions

Given: AC, the invertible probability-preserving ergodic transformation $T$ , centered real $X_{0} \in L^{2} (Q)$ , its stationary translates, natural past filtration, and the displayed summability assumption.

[A1]

AC supplies the inherited choices for the following conditional-expectation and limit suppliers. (The Axiom of Choice)

[F1]

Conditional expectation is linear, preserves expectation and constants, and contracts the real $L^{2}$ norm. (Basic algebra and order properties of conditional expectation, Conditional lp contraction)

[F2]

Conditional expectations satisfy the tower property and allow a known measurable factor to be taken out when the input and the required products are integrable. (Tower property of conditional expectation, Taking out what is known)

[F3]

Under countable choice $L^{2}$ is complete. (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ )

[F4]

Square-integrable products satisfy Cauchy–Schwarz. (Cauchy-Schwarz for random variables)

[F5]

A stationary integrable process with ergodic canonical shift has averages converging almost surely to its expectation. Ergodicity requires invariant events to have probability zero or one. (Birkhoff limit for a stationary integrable process, Ergodicity relative to an invariant measure)

[F6]

A square-integrable martingale-difference array with terminal conditional variance clock tending to $1$ in probability and vanishing unconditional Lindeberg sums has terminal sum converging to $N (0, 1)$ . (Martingale central limit theorem)

[F7]

Dominated convergence passes limits through integrals dominated by an integrable function. (Dominated convergence)

[F8]

$L^{2}$ convergence implies convergence in probability; a perturbation tending to zero in probability preserves a weak limit; continuous scaling preserves weak convergence. ( $L^{p}$ convergence implies convergence in probability, Converging together lemma, Continuous mapping theorem)

[F9]

Variance and covariance are centered second and product moments. (Moments, variance, and covariance on a probability space)

Proof

1.1A1F1given

For an integrable $Z$ and any integer $k$ , the variable $E [Z ∣ F_{0}] \circ T^{k}$ is $F_{k}$ measurable because the pullback of $F_{0}$ by $T^{k}$ is $F_{k}$ . Every event in $F_{k}$ is the pullback of an event in $F_{0}$ , since $T$ is invertible and bimeasurable. Changing variables under the probability-preserving bijection $T^{k}$ proves that the integrals of this candidate and $Z \circ T^{k}$ agree over every $F_{k}$ event. Uniqueness of conditional expectation therefore gives $E [Z \circ T^{k} ∣ F_{k}] = E [Z ∣ F_{0}] \circ T^{k}$ as almost-everywhere classes.

2.1A1F3step 1.1givenconstruct

By summability and [F3], the partial sums defining $H_{0}$ are Cauchy in $L^{2} (F_{0})$ and have a limit there. Composition with $T^{k}$ is an $L^{2}$ isometry, so step 1.1 yields $H_{k} = \sum_{j \geq 1} E [X_{k + j} ∣ F_{k}]$ in $L^{2}$ , with $H_{k}$ measurable in $F_{k}$ and $∥ H_{k} ∥_{2} = ∥ H_{0} ∥_{2}$ . Representatives may be fixed for the countably many indices; discard a countable union of null sets for their identities.

3.1F1F2F8step 1.1step 2.1algebra

By [F1], conditional expectation is continuous in $L^{2}$ . Apply [F2]'s tower property to the partial sums in step 2.1 and pass to the $L^{2}$ limit to obtain $E [H_{k} ∣ F_{k - 1}] = \sum_{j \geq 1} E [X_{k + j} ∣ F_{k - 1}] = H_{k - 1} - E [X_{k} ∣ F_{k - 1}]$ . Thus $D_{k} = X_{k} + H_{k} - H_{k - 1}$ is square-integrable and $F_{k}$ measurable, with $E [D_{k} ∣ F_{k - 1}] = 0$ . Also $D_{k} = D_{1} \circ T^{k - 1}$ , so the differences are stationary. Summing their defining identity gives $S_{n} = M_{n} + R_{n}$ , where $M_{n} = \sum_{k = 1}^{n} D_{k}$ , $R_{n} = H_{0} - H_{n}$ , and $∥ R_{n} ∥_{2} \leq 2 ∥ H_{0} ∥_{2}$ . In particular $R_{n} / \sqrt{n} \to 0$ in $L^{2}$ and hence in probability by [F8].

4.1F1F5step 1.1step 3.1given

Put $q_{0} = E [D_{1}^{2} ∣ F_{0}]$ . It is nonnegative, integrable, and has expectation $τ^{2} = E [D_{1}^{2}]$ by [F1]. Step 1.1 gives $E [D_{k}^{2} ∣ F_{k - 1}] = q_{0} \circ T^{k - 1}$ . The map $ω \mapsto (q_{0} (T^{j} ω))_{j \geq 0}$ intertwines $T$ with the canonical one-sided shift. Hence a shift-invariant event for this process pulls back to a $T$ -invariant event modulo null sets, whose probability is zero or one by the assumed ergodicity. The process is stationary, integrable and ergodic, so [F5] gives $n^{- 1} \sum_{k = 1}^{n} E [D_{k}^{2} ∣ F_{k - 1}] \to τ^{2}$ almost surely. The limiting variance is therefore a constant.

5.1A1F6F7F8step 3.1step 4.1construct

If $τ^{2} > 0$ , let $Z_{n, k} = D_{k} / (τ \sqrt{n})$ for $1 \leq k \leq n$ , and let $Z_{n, k} = 0$ for $k > n$ ; use the filtration $F_{n, k} = F_{\min (k, n)}$ for $k \geq 0$ . Step 3.1 makes each row a square-integrable martingale-difference row. The row sums and clocks are eventually constant, and step 4.1 shows their terminal clocks converge almost surely to $1$ . For every $ε > 0$ , stationarity gives $\sum_{k \geq 1} E [Z_{n, k}^{2} 1_{{∣ Z_{n, k} ∣ > ε}}] = τ^{- 2} E [D_{1}^{2} 1_{{∣ D_{1} ∣ > ε τ \sqrt{n}}}] \to 0$ by [F7]. Thus [F6] gives $M_{n} / (τ \sqrt{n}) \Rightarrow N (0, 1)$ . Scaling and the perturbation in step 3.1 give $S_{n} / \sqrt{n} \Rightarrow N (0, τ^{2})$ by [F8].

6.1F8step 3.1step 5.1given

If $τ^{2} = 0$ , stationarity implies $E [D_{k}^{2}] = 0$ for every $k$ , so $M_{n} = 0$ almost surely. Step 3.1 gives $S_{n} / \sqrt{n} \to 0$ in $L^{2}$ and hence in probability. The constant sequence $0$ has weak limit the point mass $N (0, 0)$ ; [F8] transfers that limit to $S_{n} / \sqrt{n}$ . This establishes the asserted CLT in both variance cases.

7.1F2F4F7F9step 3.1given∎

Suppose the additional covariance summability holds, and put $γ_{j} = Cov (X_{0}, X_{j})$ . Stationarity and expansion of the finite square give $Var (S_{n}) / n = γ_{0} + 2 \sum_{j = 1}^{n - 1} (1 - j / n) γ_{j} \to γ_{0} + 2 \sum_{j \geq 1} γ_{j}$ , by [F7] on counting measure with summable dominator $∣ γ_{j} ∣$ . For $i < j$ , $D_{i}$ is $F_{j - 1}$ measurable, so [F2], [F4], and step 3.1 give $E [D_{i} D_{j}] = 0$ . Therefore $E [M_{n}^{2}] = n τ^{2}$ . The identity $S_{n} = M_{n} + R_{n}$ and [F4] imply $∣ E [S_{n}^{2}] - n τ^{2} ∣ / n \leq ∥ R_{n} ∥_{2}^{2} / n + 2 ∥ R_{n} ∥_{2} τ / \sqrt{n} \to 0$ , since $∥ R_{n} ∥_{2}$ is bounded. All $S_{n}$ are centered, so [F9] identifies the two variance limits and proves the stated covariance formula, including $τ^{2} = 0$ .

Source notes

The absolutely summable projection assumption permits an exact $L^{2}$ coboundary series, so no maximal martingale approximation theorem is imported. The deterministic variance follows from the explicitly assumed ergodicity and its factor argument. The covariance identity is obtained from second moments, independently of convergence in distribution.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Finite moments control increments over short moving windows

Statement

Fix $q > p > 2$ . Let $(Z_{k})_{k \geq 1}$ be identically distributed centered real random variables with $E ∣ Z_{1} ∣^{q} < \infty$ . Assume either that they are independent, or that they are one-dependent, meaning that $σ (Z_{1}, \dots, Z_{k})$ and $σ (Z_{k + 2}, Z_{k + 3}, \dots)$ are independent for every $k \geq 1$ . Put $S_{0} = 0$ and $S_{j} = \sum_{k = 1}^{j} Z_{k}$ .

There is an event of probability one on which, for every $A > 0$ and $d \geq 0$ , with $r_{n} = ⌈ A n^{1 / p} (\log n)^{d} ⌉$ for $n \geq 2$ , $\max_{\begin{array}{c} 0 \leq i < j \leq n \\ j - i \leq r_{n} \end{array}} ∣ S_{j} - S_{i} ∣ = o (n^{1 / p}) .$

Facts & Assumptions

Given: $q > p > 2$ , the centered sequence, its common finite $q$ th absolute moment, and one of the two dependence hypotheses in the Statement.

[F1]

Independent centered real variables bounded by $b$ satisfy the two-sided Bernstein bound $P (∣ \sum_{i} X_{i} ∣ \geq x) \leq 2 \exp (- x^{2} / [2 (v + b x / 3)])$ when the sum of their variances $v$ is positive; when $v = 0$ the sum is zero almost surely. Bernstein inequality for bounded independent sums

[F2]

For a nonnegative random variable $Y$ and $t > 0$ , $P (Y \geq t) \leq E Y / t$ . Markov's inequality for random variables

[F3]

If the sum of event probabilities is finite, only finitely many of the events occur almost surely; independence is unnecessary. First Borel-Cantelli lemma for events

[F4]

Independence is independence of the generated sigma-algebras; iid means independence together with equality of marginal laws. Independent random elements, Identical distribution and IID families

[F5]

For centered square-integrable $Z$ , $Var (Z) = E Z^{2}$ . Moments, variance, and covariance on a probability space

Proof

1.1givenF2F5choose

First assume independence and fix positive integers $A$ and $d$ with $d \geq 0$ . Choose $a$ strictly between $p$ and $q$ , put $M_{q} = E ∣ Z_{1} ∣^{q}$ , and, for dyadic integers $N = 2^{m} \geq 2$ , put $b_{N} = N^{1 / a}$ , $ℓ_{N} = \log N$ , and $r_{N} = ⌈ A N^{1 / p} ℓ_{N}^{d} ⌉$ . The second moment $v = E Z_{1}^{2}$ is finite since $∣ z ∣^{2} \leq 1 + ∣ z ∣^{q}$ . If $v = 0$ , the nonnegative variable $Z_{1}^{2}$ has zero expectation and is zero almost surely: for each integer $h \geq 1$ , [F2] bounds $P (Z_{1}^{2} \geq 1 / h)$ by zero, and the union of these events is ${Z_{1} \neq 0}$ . Equality of marginal laws and countable subadditivity make every $Z_{k}$ zero simultaneously almost surely. This proves the assertion in this case. Henceforth assume $v > 0$ .

2.1givenF2F3F4F5step 1.1

The union bound and [F2] give $P (\max_{k \leq N} ∣ Z_{k} ∣ > b_{N}) \leq M_{q} N^{1 - q / a}$ . Since $1 - q / a < 0$ , these probabilities are summable over dyadic $N$ . By [F3], eventually no one of the first $N$ variables is truncated at level $b_{N}$ , almost surely along the dyadic horizons. Define $m_{N} = E [Z_{1} 1_{{∣ Z_{1} ∣ \leq b_{N}}}]$ and $V_{N, k} = Z_{k} 1_{{∣ Z_{k} ∣ \leq b_{N}}} - m_{N}$ . For each fixed $N$ , these variables are independent and centered, $∣ V_{N, k} ∣ \leq 2 b_{N}$ , and $Var (V_{N, k}) \leq v$ . Moreover centering of $Z_{1}$ gives $∣ m_{N} ∣ \leq E [∣ Z_{1} ∣ 1_{{∣ Z_{1} ∣ > b_{N}}}] \leq M_{q} b_{N}^{1 - q}$ .

3.1F1F3step 1.1step 2.1algebra

For an interval of at most $r_{N}$ indices, [F1] bounds the tail of its $V_{N, k}$ sum by $2 \exp (- x^{2} / [2 (v r_{N} + 2 b_{N} x / 3)])$ ; if its actual variance is zero the bound still holds because the tail is zero. Set $x_{N} = C (\sqrt{r_{N} ℓ_{N}} + b_{N} ℓ_{N})$ , where $C \geq \max {\sqrt{20 v}, 40 / 3, 1}$ . Then $2 v r_{N} \leq x_{N}^{2} / (10 ℓ_{N})$ and $4 b_{N} x_{N} / 3 \leq x_{N}^{2} / (10 ℓ_{N})$ , so the bound is at most $2 e^{- 5 ℓ_{N}} = 2 N^{- 5}$ . There are at most $N^{2}$ intervals with integer endpoints in $[0, N]$ . The probability that any interval of length at most $r_{N}$ violates the bound $x_{N}$ is therefore at most $2 N^{- 3}$ , summable over dyadic $N$ . By [F3], all such interval sums are bounded by $x_{N}$ eventually almost surely.

4.1step 1.1step 2.1step 3.1algebra

On the eventual no-truncation event from step 2.1, every original interval sum differs from its centered truncated sum by at most $r_{N} ∣ m_{N} ∣$ . Now $\sqrt{r_{N} ℓ_{N}} = o (N^{1 / p})$ because $1 / (2 p) < 1 / p$ , and $b_{N} ℓ_{N} = o (N^{1 / p})$ because $1 / a < 1 / p$ . Also $r_{N} ∣ m_{N} ∣ \leq M_{q} r_{N} N^{(1 - q) / a} = o (N^{1 / p})$ . Thus step 3.1 proves the claimed little-o bound at dyadic horizons. These comparisons follow by dividing by $N^{1 / p}$ : in each expression a strictly negative power of $N$ multiplies a fixed power of $\log N$ , which tends to zero.

5.1F3step 4.1choose

For $2^{m - 1} < n \leq 2^{m} = N$ , monotonicity of $n^{1 / p} (\log n)^{d}$ gives $r_{n} \leq r_{N}$ , so every admissible interval for $n$ is included among those for $N$ . Since $N^{1 / p} \leq 2^{1 / p} n^{1 / p}$ , step 4.1 proves the assertion at every horizon $n$ . Intersect the probability-one events obtained for the countably many integer pairs $A \geq 1, d \geq 0$ . For arbitrary real $A > 0, d \geq 0$ , choose integers $A^{'} \geq A$ and $d^{'} \geq d$ ; when $n \geq 3$ , $\log n \geq 1$ makes the real-parameter window no longer than the integer-parameter window. The conclusion therefore holds on one event for every $A, d$ in the Statement.

6.1givenF4step 1.1step 5.1algebra∎

Under one-dependence, each parity subsequence is independent. Indeed, the sigma-algebra of any finite initial collection of even variables is contained in $σ (Z_{1}, \dots, Z_{2 h})$ , which is independent of $Z_{2 h + 2}$ by the given separation; induction on the number of factors proves the finite rectangle factorization for all even variables. The identical argument applies to the odd variables. Both subsequences retain the common centered marginal and its $q$ th moment. Apply step 5.1 to each subsequence and intersect their two probability-one events. An interval of at most $r_{n}$ original indices splits into one contiguous interval of each parity, each with at most $r_{n}$ terms and with its subsequence endpoints at most $n$ . Each component is $o (n^{1 / p})$ by the independent result, uniformly over the original intervals; their sum has the same bound. No independence between the parity subsequences is asserted or needed. All windows are nonempty by $i < j$ ; empty component windows contribute zero. This completes both dependence cases, including zero variance and every permitted window parameter.

Remarks

The estimate also controls short calendar-time reward increments once a genuine cycle decomposition is supplied. Suppose cycle lengths are positive integers, centered complete-cycle rewards satisfy the lemma, and each partial-cycle reward is bounded by an identically distributed nonnegative envelope $H_{k}$ with $E H_{1}^{q} < \infty$ . Dyadic Markov bounds as in step 1.2, at levels $N^{1 / a}$ with $p < a < q$ , give $\max_{k \leq N + 2} H_{k} = o (N^{1 / p})$ almost surely. An interval of at most $r_{n}$ calendar steps crosses at most $r_{n} + 2$ complete or partial cycles. Its full-cycle contribution is controlled by the lemma and its two endpoint pieces by twice the maximum envelope. Thus its centered reward increment is $o (n^{1 / p})$ uniformly up to time $n$ . This remark is conditional on the actual decomposition and its moment/dependence hypotheses; it asserts no regeneration theorem or Brownian coupling.

The cited full source's Lemma 5 was inspected. Its proof uses Nagaev's inequality for a probability estimate. The proof here instead uses the already proved local Bernstein supplier and the spare moment $q > p$ ; no unproved Nagaev or strong-approximation theorem is used.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Independent Brownian motion straightens a Poisson clock

Statement

Assume AC. Let $G$ be a sigma-algebra on a probability space, let $λ > 0$ , and let $N = (N (t))_{t \geq 0}$ be $G$ -measurable at each time. Suppose $N (0) = 0$ , its paths are nondecreasing right-continuous integer-valued functions finite on bounded intervals, its increments on disjoint time intervals are independent, and $N (t) - N (s)$ has law $Pois (λ (t - s))$ for $0 \leq s < t$ . Let $B$ be a standard Brownian motion independent of $G$ .

On an extension preserving the original probability space there is a standard Brownian motion $W$ , independent of $G$ , such that $\sup_{0 \leq t \leq n} ∣ B (t) - λ^{- 1 / 2} W (N (t)) ∣ = O ((\log (n + 2))^{2}) almost surely as n \to \infty .$ The almost-sure bound has a finite random constant and a finite random starting index. In particular $W$ is independent of the entire Poisson clock; $W$ and $B$ are not asserted to be independent.

Facts & Assumptions

Given: AC, $G$ , $λ > 0$ , $N$ , and the independent Brownian motion $B$ satisfying the Statement.

[A1]

AC is assumed. The Axiom of Choice

[F1]

Brownian motion is continuous on a common probability-one event and is centered Gaussian with covariance $\min (s, t)$ ; conversely that covariance with common path continuity gives Brownian motion. Brownian motion, Brownian covariance is equivalent to independent stationary normal increments

[F2]

Finite jointly Gaussian blocks with zero cross-covariance are independent, including singular blocks. Uncorrelated jointly Gaussian blocks are independent

[F3]

A standard Brownian motion exists, and the countable product construction adjoins countably many independent copies independently of the original space. Existence of continuous Brownian motion, Assuming countable and dependent choice, countable products of arbitrary probability spaces, Coordinate random elements of a countable product are independent

[F4]

The Poisson mass at $k \geq 0$ with parameter $a > 0$ is $e^{- a} a^{k} / k!$ ; at parameter zero it is the unit mass at zero. The exponential is its convergent power series and satisfies $e^{a + b} = e^{a} e^{b}$ . The scalar Poisson count law, The real exponential function and the number $e$ by a power series, The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$

[F5]

For a standard normal $G$ and $x > 0$ , $P (G > x) \leq (2 π)^{- 1 / 2} x^{- 1} e^{- x^{2} / 2}$ ; reflection gives $P (\sup_{s \leq h} U (s) > x) = 2 P (U (h) > x)$ for Brownian $U$ , $h > 0$ . Two-sided Mills bounds for the standard normal tail, Law of the Brownian maximum

[F6]

Markov's inequality applies to nonnegative variables; summable event probabilities imply that only finitely many events occur almost surely. Markov's inequality for random variables, First Borel-Cantelli lemma for events

[F7]

A differentiable map with uniformly bounded derivative on a convex open set is Lipschitz there. On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$

[F8]

Agreement of finite measures on a generating pi-system extends to its sigma-algebra. Dynkin's pi-lambda theorem

Proof

1.1givenA1F3F4

Work on common full-measure events of Brownian continuity and the assumed clock paths. For each integer $K \geq 0$ , [F4] gives $P (N (n) \leq K) = e^{- λ n} \sum_{k = 0}^{K} (λ n)^{k} / k! \to 0$ . The events ${N (n) \leq K}$ decrease in $n$ , so continuity of probability shows that $N (n)$ is unbounded almost surely after taking the countable union over $K$ . By [A1] and [F3], extend the space by independent standard Brownian motions $(U_{j})_{j \geq 0}$ , independent also of the original variables. They will supply continuous bridges; the original space and all its joint laws are preserved by this product extension.

1.2givenF4F6algebra

We prove the necessary uniform clock bounds directly. If $X \sim Pois (a)$ , summing [F4]'s nonnegative series gives $E e^{θ (X - a)} = \exp (a (e^{θ} - 1 - θ))$ for every real $θ$ . For $∣ θ ∣ < 3$ , the exponential series and $k! \geq 2 \cdot 3^{k - 2}$ for $k \geq 2$ imply $e^{θ} - 1 - θ \leq θ^{2} / [2 (1 - ∣ θ ∣ / 3)]$ . Exponential Markov and the choice $θ = x / (a + x / 3)$ , applied with either sign, therefore give $P (∣ X - a ∣ \geq x) \leq 2 \exp (- x^{2} / [2 (a + x / 3)])$ when $a > 0, x > 0$ ; at $a = 0$ the deviation is zero. With $a = λ (v - u)$ this implies $P (∣ N (v) - N (u) - λ (v - u) ∣ \geq x) \leq 2 \exp (- c_{λ} \min {x^{2} / (v - u), x})$ for a constant $c_{λ} > 0$ and integer $0 \leq u < v$ . This derivation uses only the Poisson mass series and Markov, not a strong approximation theorem.

2.1givenF1F2step 1.1construct

Partition the original time axis into macrointervals $[0, 1], [1, 2], [2, 4], [4, 8], \dots$ . For each macrointerval $I = [a, a + D]$ , extract the root normal $R_{I} = (B (a + D) - B (a)) / \sqrt{D}$ . For every dyadic cell $J = [b, b + d]$ inside a macrointerval, of length $d \geq 2$ , extract its midpoint normal $G_{J} = 2 (B (b + d / 2) - (B (b) + B (b + d)) / 2) / \sqrt{d}$ . This is a countable iid standard-normal family, independent of $G$ . To verify independence, the midpoint contrast on $J$ has covariance zero with the endpoint increment on $J$ and with all Brownian evaluations outside its interior: for $t \leq b$ the three covariance terms cancel, and for $t \geq b + d$ they are $2 (b + d / 2) - b - (b + d) = 0$ . Its variance is one by direct expansion of $\min (s, t)$ . Two dyadic cells are either disjoint or nested; in the nested case, the larger cell's contrast uses only evaluations outside the smaller cell's interior or at its endpoints. Hence distinct contrasts and roots have zero covariance. Every finite family is jointly Gaussian by [F1], so repeated use of [F2] proves mutual independence.

2.2F6step 1.2algebrachoose

For dyadic horizons $n = 2^{m}$ , write $ℓ = \log (n + 2)$ . Applying step 1.2 with $x = K (\sqrt{(v - u) ℓ} + ℓ)$ and choosing a sufficiently large deterministic $K$ makes the probability at most $2 (n + 2)^{- 6}$ . There are at most $(n + 1)^{2}$ integer intervals in $[0, n]$ . The union bound and [F6] show that, almost surely eventually on dyadic horizons, every such interval satisfies $∣ N (v) - N (u) - λ (v - u) ∣ \leq K (\sqrt{(v - u) ℓ} + ℓ)$ . Choose a deterministic $A > 1$ sufficiently large. For every dyadic cell of length $d \geq A ℓ$ , its normalized half-counts $α = L_{J} / (λ d)$ and $β = Q_{J} / (λ d)$ then lie in $[1 / 4, 3 / 4]$ . For every integer point $j$ in the cell, with $x = (j - b) / d$ and $z = (N (j) - N (b)) / (λ d)$ , one has $∣ z - x ∣ + ∣ α - 1 / 2 ∣ + ∣ β - 1 / 2 ∣ \leq C \sqrt{ℓ / d}$ . Indeed the interval estimate gives this bound first with an additional term $C ℓ / d$ , which is at most another constant times $\sqrt{ℓ / d}$ since $d \geq A ℓ$ . The same estimate implies that every integer interval of length at most $2 A ℓ$ has count increase at most $C ℓ$ . Constants here and below are deterministic and may depend on $λ, A, K$ .

3.1F1F3step 1.1step 2.1construct

For a cell $J = [b, b + d]$ put $C_{J} = N (b + d) - N (b)$ , with half-counts $L_{J} = N (b + d / 2) - N (b)$ and $Q_{J} = C_{J} - L_{J}$ when $d \geq 2$ . Prescribe a $W$ increment $\sqrt{C_{I}} R_{I}$ over the count interval $[N (a), N (a + D)]$ of each macrointerval, starting from $W (0) = 0$ and concatenating the macrointervals. Given both endpoint values for $J$ , if $C_{J} > 0$ prescribe its count-midpoint value as $W (N (b)) + (L_{J} / C_{J}) (W (N (b + d)) - W (N (b))) + \sqrt{L_{J} Q_{J} / C_{J}} G_{J}$ . When $C_{J} = 0$ all three count times agree and the midpoint value is that endpoint. When only one half-count is zero the displayed prescription also equals the appropriate existing endpoint. Recurse down to unit cells. For a unit cell $[j, j + 1]$ of count length $c > 0$ , with left count time $h = N (j)$ and prescribed endpoint increment $Y$ , fill its count interval continuously by $W (h + s) = W (h) + (s / c) Y + \sqrt{c} (U_{j} (s / c) - (s / c) U_{j} (1))$ for $0 \leq s \leq c$ . Zero-count leaves require no filling. Unboundedness from step 1.1 makes these intervals cover $[0, \infty)$ . All operations depend measurably on finitely many integer counts and Gaussian evaluations for each fixed $W (t)$ : partition by the first integer $j$ with $N (j) > t$ and its finite count history. The resulting countable pieces give measurable evaluations, including endpoints.

3.2F1F5F6step 2.1

The tail bound in [F5] gives $P (∣ G ∣ > x) \leq 2 e^{- x^{2} / 2}$ for $x \geq 1$ . On $[0, n]$ the number of root and midpoint coefficients in step 2.1 is at most $2 n$ . A union bound with $x = K_{1} \sqrt{ℓ}$ , followed by [F6] with $K_{1}$ sufficiently large, shows that their maximum absolute value is $O (\sqrt{ℓ})$ almost surely eventually along dyadic horizons. Also [F5] and the union bound give, for Brownian $U$ , $P (\sup_{s \leq h} ∣ U (s) ∣ > x) \leq 4 e^{- x^{2} / (2 h)}$ when $x / \sqrt{h} \geq 1$ . Therefore a bridge $U (s) - (s / h) U (h)$ over length $h$ has supremum tail bounded by $4 e^{- x^{2} / (8 h)}$ for $x / (2 \sqrt{h}) \geq 1$ , since its supremum is at most twice that of $U$ . These estimates also apply to translated Brownian increment paths by [F1].

4.1F1F2F3step 2.1step 3.1algebra

Fix a deterministic finite count history. The Gaussian split used in step 3.1 is exact: if an interval has total increment $Y \sim N (0, c)$ and child lengths $l, r \geq 0$ , $l + r = c > 0$ , independent $G \sim N (0, 1)$ gives child increments $Y_{L} = (l / c) Y + \sqrt{l r / c} G$ and $Y_{R} = Y - Y_{L}$ . Their variances are $l, r$ and their covariance is $l r / c - l r / c = 0$ . Thus they are independent by [F2], with zero-length increments identically zero. Recursing proves independent normal increments of the prescribed count lengths over all nonzero unit leaves, also across macrointervals. For the independent bridge $β (s) = U_{j} (s) - s U_{j} (1)$ , [F1] directly gives mean zero, continuous paths, endpoints zero, and covariance $\min (s, t) - s t$ . Filling a leaf as in step 3.1 therefore gives centered Gaussian increments whose covariance at offsets $s, t$ is $s t / c + c (\min (s / c, t / c) - s t / c^{2}) = \min (s, t)$ . Different leaves have independent increment processes because their endpoint increments are independent and all bridge motions are independent. Summing these increments shows that, for every finite evaluation list, $W$ has centered Gaussian covariance $\min (s, t)$ on this fixed count history.

4.2step 2.1step 3.1step 2.2algebra

Fix a large dyadic horizon and a macrointerval $I = [a, a + D] \subseteq [0, n]$ . If $D > A ℓ$ , stop its binary tree at cells with lengths in $[A ℓ, 2 A ℓ)$ ; if $D \leq A ℓ$ , stop at the root. For an integer $j \in I$ , consider the piecewise-linear interpolant of $B (j) - B (a)$ on this stopped grid, and the piecewise-linear interpolant of $λ^{- 1 / 2} (W (N (j)) - W (N (a)))$ on the corresponding count grid, evaluated at $N (j)$ . Repeated midpoint interpolation from step 3.1 expands each interpolant in the same root normal and in the midpoint normals of its ancestor chain. In an active cell of length $d$ containing $j$ in its left half, the coefficients of its midpoint normal are respectively $\sqrt{d} x$ and $\sqrt{d} z \sqrt{β / [α (α + β)]}$ , with $x, z, α, β$ as in step 2.2. This follows by multiplying the left interpolation fraction $z / α$ by $λ^{- 1 / 2} \sqrt{L_{J} Q_{J} / C_{J}}$ . In the right half the coefficients are obtained by replacing $(x, z, α, β)$ by $(1 - x, α + β - z, β, α)$ . At a midpoint or endpoint these same formulas agree with the endpoint prescription.

5.1F1F3F8step 1.1step 3.1step 4.1

For any finite list of evaluation times, partition the original space into countably many $G$ -events specifying a long enough finite integer count history to cover them. The Gaussian and added bridge variables are independent of $G$ ; step 4.1 shows that on every such count configuration the resulting evaluation law is the same Brownian Gaussian law. Consequently, for any $A \in G$ and any Borel set $H$ of that finite evaluation vector, summing over this partition gives $P (A \cap {(W (t_{i}))_{i} \in H}) = P (A) ν_{H}$ , where $ν_{H}$ is its Brownian probability. No regular conditional probability on the original space is needed. The construction has common continuous paths: count intervals have integer endpoints, finitely many nonzero intervals meet each bounded count-time interval, their bridges are continuous, and endpoint prescriptions agree. Thus [F1] makes $W$ standard Brownian. Continuous evaluations are limits of rational evaluations, and [F8] extends the factorization from rational cylinders to $σ (W (t) : t \geq 0)$ . Therefore $W$ is independent of all of $G$ , not just of $N$ .

5.2F7step 2.2step 3.2step 4.2algebra

The function $z \sqrt{β / [α (α + β)]}$ has uniformly bounded first derivatives for $1 / 4 \leq α, β \leq 3 / 4$ and $0 \leq z \leq 2$ : its displayed denominators are bounded away from zero, and differentiating gives only bounded products of these reciprocals and square roots. One may take a slightly larger open convex box still with positive $α, β$ and use [F7]. At $(α, β, z) = (1 / 2, 1 / 2, x)$ its value is $x$ . Hence step 2.2 bounds the midpoint coefficient discrepancy by $C \sqrt{ℓ}$ . The normalized right-half prefix differs from its deterministic value by at most another constant times $\sqrt{ℓ / d}$ , so the same bound applies there. The root coefficients for a large macrointerval are $\sqrt{D} x$ and $\sqrt{D} z / \sqrt{C_{I} / (λ D)}$ ; the same bounded-derivative argument and step 2.2 bound their discrepancy by $C \sqrt{ℓ}$ . By step 3.2, each shared Gaussian coefficient therefore contributes at most $C ℓ$ to the interpolant difference. At most $1 + \log_{2} n$ coefficients occur on an ancestor chain, so the increment-interpolant error is $O (ℓ^{2})$ .

6.1F6step 4.1step 5.1step 2.2step 3.2step 4.2

We control the discarded bridge pieces uniformly. Each stopped cell has deterministic length at most $2 A ℓ$ and, on the count event from step 2.2, count length at most $C ℓ$ . The Brownian deviation from linear interpolation on a cell is a continuous bridge with the covariance computed in step 4.1, regardless of the endpoint values. Step 3.2 therefore bounds the probability that its supremum exceeds $K_{2} ℓ$ by $4 e^{- c K_{2}^{2} ℓ}$ . There are at most $n$ stopped cells. For the $W$ bridges the same estimate holds on every count history having these length bounds, by the conditional Gaussian calculation in steps 4.1, 5.1; summing over the count configurations gives the identical unconditional bound on the good count event. Choose $K_{2}$ large, add the summable failure probability of that count event, and apply [F6]. Thus all discarded $B$ and scaled $W$ bridge pieces are $O (ℓ)$ almost surely eventually. This argument includes zero count length, whose bridge is identically zero.

7.1step 2.2step 3.2step 5.2step 6.1algebra

A completed large macrointerval contributes only its root error $∣ \sqrt{D} - \sqrt{C_{I} / λ} ∣ ∣ R_{I} ∣ \leq C ℓ$ by steps 2.2, 3.2. For a macrointerval of length at most $A ℓ$ , its entire $B$ increment supremum and its scaled $W$ increment supremum are $O (ℓ)$ by the same reflection and count-history argument as step 6.1, applied without subtracting an interpolant. There are at most $1 + \log_{2} n$ macrointervals before time $n$ . Add the root or small-macrointerval errors of the completed macrointervals, then the interpolant error from step 5.2 and the two leaf errors from step 6.1 in the last one. This gives $\max_{j \leq n, j \in N_{0}} ∣ B (j) - λ^{- 1 / 2} W (N (j)) ∣ = O (ℓ^{2})$ almost surely along dyadic horizons.

8.1givenF1F5F6step 5.1step 2.2step 3.2step 7.1

For real $t \in [j, j + 1]$ , Brownian reflection and a union bound over the $n$ unit intervals make $∣ B (t) - B (j) ∣ = O (\sqrt{ℓ})$ uniformly eventually almost surely. Step 2.2 gives $N (j + 1) - N (j) \leq C ℓ$ . On every such count history, step 5.1 gives Brownian law for $W$ , and reflection bounds its increment supremum on each $[N (j), N (j + 1)]$ by $K_{3} ℓ$ except on a probability bounded by $4 n e^{- c K_{3}^{2} ℓ}$ ; this is summable on dyadic horizons when $K_{3}$ is large. Integrating over the count configurations and using [F6] shows that these $W$ increments are uniformly $O (ℓ)$ . Monotonicity of $N$ places $N (t)$ in that count interval. Add these two bounds to step 7.1 to obtain the required real-time supremum on dyadic horizons. For any real $n \geq 2$ , the next dyadic horizon is less than $2 n$ , and its logarithm differs from $\log (n + 2)$ by a bounded additive constant, proving the asserted bound at every horizon.

9.1A1F1F2F3step 1.1step 3.1step 4.1step 5.1step 7.1step 8.1∎

The finite initial time range is absorbed into the random constant: $B$ is bounded there by continuity, $N$ is bounded there by hypothesis, and $W$ is bounded on the corresponding compact count-time interval by continuity. Repeated or zero count times were treated in steps 3.1, 4.1. The restriction $λ > 0$ is needed for the displayed scaling and the lower large-cell count bounds; no assertion is made at rate zero. Step 5.1 proves the full claimed independence, while steps 7.1, 8.1 prove the approximation. AC was used precisely through [F1]–[F3] and the product extension in step 1.1; all concentration, interpolation, and countable-event arguments are local.

Source notes

The complete source PDF was retrieved. Lemma 2.4's full statement, its auxiliary-randomness clause, and the opening Haar construction were inspected, together with the introduction and proof roadmap. The source's detailed exponential-tail estimates were not represented as independently verified. The proof above supplies its own weaker logarithmic-square bound, which is sufficient for every fixed positive polynomial approximation scale. It invokes no KMT, embedding, renewal approximation, or unproved strong-approximation result.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Gaussian anchored bounded sums have uniform cumulant and Fourier bounds

Statement

Assume AC. Fix $0 < γ \leq 1$ . Let $1 \leq d \leq 4$ , $s > 0$ , $B > 0$ , and let $X_{1}, \dots, X_{m}, Z$ be independent centered random vectors in $R^{d}$ , where $m \geq 0$ , $∣ X_{j} ∣ \leq B$ almost surely, and $Z \sim N_{d} (0, G)$ . Suppose $S = Z + \sum_{j} X_{j}$ has covariance $s I_{d}$ and $G \geq γ s I_{d}$ in quadratic form order. There is an absolute constant $C_{0} \geq 1$ such that, with $τ = C_{0} B$ , the logarithmic moment generating function $K (z) = \log E e^{z \cdot S}, K (0) = 0,$ has a single analytic branch on the complex Euclidean ball $τ ∣ z ∣ < 1$ , and $∣ \partial_{u} \partial_{v}^{2} K (z) ∣ \leq τ s ∣ u ∣ ∣ v ∣^{2} (u, v \in R^{d}, τ ∣ z ∣ < 1) .$ Here $z \cdot S$ uses the complex bilinear dot product and $∣ z ∣^{2} = \sum_{i} ∣ z_{i} ∣^{2}$ . Every real exponential tilt $P_{h} (d ω) = e^{h \cdot S - K (h)} P (d ω)$ is defined, and its characteristic function satisfies $∣ φ_{h} (t) ∣ : = ∣ E_{h} e^{i t \cdot S} ∣ \leq e^{- γ s ∣ t ∣^{2} / 2} (t, h \in R^{d}),$ where $K (h) = \log E e^{h \cdot S}$ for real $h$ , also outside the analytic ball. Put $κ = τ / \sqrt{s}$ . For every fixed $a > 0$ and integer $N \geq 1$ there is a finite $C_{a, N, γ}$ , uniform in $1 \leq d \leq 4$ , such that, if $0 < κ \leq 1$ , $(2 π)^{- d} \int_{∣ t ∣ \geq a / τ} ∣ φ_{h} (t) ∣ d t \leq C_{a, N, γ} κ^{N} s^{- d / 2},$ $(2 π)^{- d} \int_{∣ t ∣ \geq a / τ} ∣ t \cdot v ∣ ∣ φ_{h} (t) ∣ d t \leq C_{a, N, γ} κ^{N} ∣ v ∣ s^{- (d + 1) / 2} .$ All assertions hold with the same $C_{0}$ , $B$ , $s$ , and $γ$ for $A S$ , where $A$ is any real $k \times d$ matrix with $1 \leq k \leq d$ and $A A^{T} = I_{k}$ . This includes every coordinate prefix, every orthogonal change of coordinates, and their compositions. In particular it covers the normalized joint sum and difference of two independent children by applying the assertion in their joint dimension, provided that dimension is at most four and the joint input satisfies the stated covariance assumptions.

Facts & Assumptions

Given: The hypotheses of the Statement; all vector norms are Euclidean, and all matrix orders are orders of real quadratic forms.

[F1]

Finite products of integrable functions of independent variables factor in expectation. Expectations factor over finite products of independent random variables.

[F2]

Dominated convergence interchanges limits and integrals under an integrable absolute majorant. Dominated convergence.

[F3]

Absolute Fubini and nonnegative Tonelli apply to the probability and Lebesgue integrals below. Fubini's theorem for L^1 functions on a sigma-finite product, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product.

[F4]

A positive definite Gaussian law has density $q_{G} (x) = (2 π)^{- d / 2} (\det G)^{- 1 / 2} e^{- x^{T} G^{- 1} x / 2}$ and a Gaussian with covariance $G$ has characteristic function $e^{- t^{T} G t / 2}$ . A positive-definite multivariate normal law has its Gaussian density, Characteristic function of a multivariate normal law.

[F6]

Locally uniformly convergent holomorphic series are holomorphic, the one-variable identity theorem extends equality from a real interval, and linear changes of Lebesgue variables have their determinant Jacobian. A locally uniformly convergent series of holomorphic functions may be differentiated term by term, Identity theorem for holomorphic functions, A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions.

[F5]

Euclidean Cauchy–Schwarz bounds $∣ u \cdot x ∣ \leq ∣ u ∣ ∣ x ∣$ , and its integral version follows by integrating $(f - c g)^{2} \geq 0$ and minimizing in real $c$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation.

[A1]

AC is assumed and inherited by the Gaussian suppliers. The Axiom of Choice.

Proof

1.1A1givenF2F5F6

Fix one centered bounded summand $X$ with covariance $D$ , and put $M (z) = E e^{z \cdot X}$ . Its exponential series converges uniformly on every bounded complex set because $∣ X ∣ \leq B$ , and the expectation of its absolute series is bounded by $e^{B ∣ z ∣}$ ; thus termwise integration defines an entire function and its derivatives are the expectations of the corresponding polynomial factors times the exponential. Fix an absolute $b > 0$ with $e^{b} - 1 \leq 1 / 2$ . If $B ∣ z ∣ < b$ , then $∣ M (z) - 1 ∣ \leq e^{B ∣ z ∣} - 1 \leq 1 / 2$ , so the convergent power series for $\log (1 + (M (z) - 1))$ supplies a single analytic logarithm vanishing at zero.

1.2F2F4F6algebra

Completing the square in the density in [F4] shows $E e^{h \cdot Z} = e^{h^{T} G h / 2}$ for real $h$ and that its real tilt is $N_{d} (G h, G)$ . The same integral for complex $z$ converges absolutely on bounded complex sets: $e^{Re (z) \cdot x}$ times the Gaussian density is a constant times a translated Gaussian density, which dominates its polynomial derivatives on smaller bounded sets. Its exponential series or differentiated integral therefore gives an entire function, and the real identity extends to complex arguments. To justify that last extension without a several-variable identity theorem, fix all but the first coordinate real and apply the one-variable identity theorem, then successively free each remaining coordinate. Thus the Gaussian cumulant is $z^{T} G z / 2$ and has zero third derivative.

2.1step 1.1F5

On this ball, writing subscripts for directional derivatives, the bounds $∣ e^{z \cdot X} ∣ \leq e^{b}$ , $∣ u \cdot X ∣ \leq B ∣ u ∣$ , and $E (v \cdot X)^{2} = v^{T} D v$ give $∣ M_{u} ∣ \leq e^{b} B ∣ u ∣$ , $∣ M_{v} ∣ \leq e^{b} \sqrt{v^{T} D v}$ , $∣ M_{v v} ∣ \leq e^{b} v^{T} D v$ , $∣ M_{u v} ∣ \leq e^{b} B ∣ u ∣ \sqrt{v^{T} D v}$ , and $∣ M_{u v v} ∣ \leq e^{b} B ∣ u ∣ v^{T} D v$ . These inequalities also hold when $D$ is singular: a direction with zero second moment has $v \cdot X = 0$ almost surely.

2.2step 1.2F1F4

Every real tilt exists because all bounded-summand moment generating functions and the Gaussian one are finite and strictly positive. Its density with respect to the original independent joint law factors as the product of the separately normalized tilt densities. Hence those summands remain independent under the tilt. Step 1.2 shows that the Gaussian factor retains covariance $G$ , while each other factor has a characteristic function of modulus at most one because $∣ e^{i t \cdot X_{j}} ∣ = 1$ . Factorization and [F4] therefore give $∣ φ_{h} (t) ∣ \leq e^{- t^{T} G t / 2} \leq e^{- γ s ∣ t ∣^{2} / 2}$ for every real $h, t$ .

3.1step 1.1step 2.1algebrachoose

Direct differentiation gives $(\log M)_{u v v} = M_{u v v} / M - (M_{u} M_{v v} + 2 M_{v} M_{u v}) / M^{2} + 2 M_{u} M_{v}^{2} / M^{3}$ . Since $∣ M ∣ \geq 1 / 2$ , step 2.1 bounds this by $C B ∣ u ∣ v^{T} D v$ for one absolute finite $C$ . Choose $C_{0}$ so that $C_{0} \geq C$ and $1 / C_{0} < b$ , and set $τ = C_{0} B$ . Each bounded summand therefore has the asserted derivative bound on $τ ∣ z ∣ < 1$ , using its own covariance.

3.2step 2.2F3F4F5F6algebra

Substitute $u = \sqrt{s} t$ in each requested tail integral, using the constant Jacobian $s^{- d / 2}$ . For $∣ u ∣ \geq a / κ$ , $e^{- γ ∣ u ∣^{2} / 2} \leq e^{- γ a^{2} / (4 κ^{2})} e^{- γ ∣ u ∣^{2} / 4}$ . For every fixed $N$ , the function $z \mapsto z^{N} e^{- γ a^{2} z^{2} / 4}$ is bounded on $z \geq 1$ by differentiation, so the extracted exponential is at most $C_{a, N, γ} κ^{N}$ . The integral of $e^{- γ ∣ u ∣^{2} / 4}$ is finite by the normal density normalization in [F4]; its product with $∣ u ∣$ is integrable as well, since $∣ u ∣ e^{- γ ∣ u ∣^{2} / 8}$ is bounded and the remaining Gaussian is integrable. Using $∣ u \cdot v ∣ \leq ∣ u ∣ ∣ v ∣$ gives the two tail bounds, with an additional $s^{- 1 / 2} ∣ v ∣$ in the weighted bound. Taking the maximum of these constants over $d = 1, 2, 3, 4$ makes them uniform.

4.1step 3.1step 1.2F1algebra

By independence [F1], the moment generating function of $S$ is the product of the factors in steps 1.1 and 1.2. On $τ ∣ z ∣ < 1$ , its logarithm is the sum of the analytic logarithms just constructed, with value zero at zero. Since the covariances add and $\sum_{j} cov (X_{j}) \leq s I_{d}$ , summing step 3.1 gives $∣ K_{u v v} (z) ∣ \leq τ ∣ u ∣ \sum_{j} v^{T} cov (X_{j}) v \leq τ s ∣ u ∣ ∣ v ∣^{2}$ . This proves the analytic assertion, including the empty bounded sum.

5.1step 4.1step 2.2step 3.2F4F5algebra∎

If $A A^{T} = I_{k}$ , the operator norm of $A$ is one: $A^{T}$ is an isometry and $∣ A x ∣ = \sup_{∣ v ∣ = 1} ∣ v \cdot A x ∣ \leq ∣ x ∣$ by [F5]. Thus $A X_{j}$ remains bounded by $B$ and centered, its sum has covariance $s A A^{T} = s I_{k}$ , and $A Z$ is Gaussian with covariance $A G A^{T} \geq γ s I_{k}$ by [F4]. Independence is preserved by applying measurable maps separately. All earlier steps therefore apply to $A S$ with the same $C_{0}$ . The map $(x, y) \mapsto ((x + y) / \sqrt{2}, (x - y) / \sqrt{2})$ is orthogonal because it preserves $∣ x ∣^{2} + ∣ y ∣^{2}$ by expansion; consequently its coordinate prefixes have the required row isometry property. This proves all assertions.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicableOpen item page →

Overlapping batch means for MCMC standard errors

Definition

Inputs: Real outputs $y_{1}, \dots, y_{n}$ , an integer $n \geq 2$ , and an integer batch size $b$ with $1 \leq b < n$ . For an MCMC run, $y_{i} = f (X_{i})$ for the chosen real-valued observable $f$ .

Outputs: The sample mean ${\overset{ˉ}{y}}_{n}$ , the overlapping-batch-means estimate ${\hat{σ}}_{O B M}^{2}$ , and its estimated Monte Carlo standard error.

Compute the sample mean ${\overset{ˉ}{y}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} y_{i} .$
For each starting index $j = 1, \dots, n - b + 1$ , compute the overlapping batch mean ${\overset{ˉ}{y}}_{j} (b) = \frac{1}{b} \sum_{i = j}^{j + b - 1} y_{i} .$
Return the overlapping-batch-means variance estimate ${\hat{σ}}_{O B M}^{2} = \frac{n b}{(n - b) (n - b + 1)} \sum_{j = 1}^{n - b + 1} ({\overset{ˉ}{y}}_{j} (b) - {\overset{ˉ}{y}}_{n})^{2} .$
Return the estimated Monte Carlo standard error $\overset{}{MCSE} ({\overset{ˉ}{y}}_{n}) = \sqrt{{\hat{σ}}_{O B M}^{2} / n} .$

All sums are finite, the denominator is positive because $b < n$ , and the variance estimate is nonnegative. The procedure always returns a number for the stated finite input. It does not by itself assert a CLT or consistency; consistency requires the hypotheses of the overlapping-batch-means theorem.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

A total-variation rate bounds strong-mixing coefficients

Statement

Assume AC. Let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with invariant probability $π$ . Suppose there are a measurable $M : X \to [0, \infty)$ with $π (M) < \infty$ and $r \in (0, 1)$ such that $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) r^{n} for every x \in X, n \geq 1.$ Then the strong-mixing coefficient of the stationary process satisfies $α_{X} (n) \leq π (M) r^{n} (n \geq 1) .$

Facts & Assumptions

Given: AC, a stationary homogeneous Markov chain with kernel $P$ and invariant law $π$ , an integrable envelope $M$ , and the displayed geometric total-variation bound.

[A1]

AC is the choice principle for the conditional classes used by the Markov-chain interfaces. (The Axiom of Choice)

[F1]

The strong-mixing coefficient is the supremum of $∣ P (A \cap B) - P (A) P (B) ∣$ over past and separated future events. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

For every bounded product-measurable future-path functional $H$ , the conditional expectation of its value on $(X_{n}, X_{n + 1}, \dots)$ is a measurable function of $X_{n}$ . (Markov property for bounded future path functionals)

[F3]

For bounded measurable $h$ , $E [h (X_{j + n}) ∣ F_{j}] = P^{n} h (X_{j})$ . (Chapman-Kolmogorov equations)

[F4]

Total variation is $∥ μ - ν ∥_{T V} = \sup_{A} ∣ μ (A) - ν (A) ∣$ . (Total variation distance for probability laws)

[F5]

A known measurable factor can be taken out of conditional expectation. (Taking out what is known)

Proof

1.1A1F2F3given

Fix $n \geq 1$ , $j \geq 0$ , $A \in F_{j}$ , and a future event $B \in σ (X_{j + n}, X_{j + n + 1}, \dots)$ . By [F2], for the path indicator defining $B$ there is a measurable $h : X \to [0, 1]$ with $E [1_{B} ∣ F_{j + n}] = h (X_{j + n})$ . Applying [F3] and the tower property gives $E [1_{B} ∣ F_{j}] = P^{n} h (X_{j})$ . Stationarity gives $P (B) = π (h)$ .

1.2F4construct

For probability measures $μ, ν$ and measurable $0 \leq h \leq 1$ , the dyadic simple functions $h_{m} = 2^{- m} \sum_{j = 1}^{2^{m}} 1_{{h \geq j 2^{- m}}}$ differ from $h$ by at most $2^{- m}$ . Their integrals differ by at most $∥ μ - ν ∥_{T V}$ , since each summand is an indicator and [F4] bounds its measure difference. Letting $m \to \infty$ proves $∣ μ (h) - ν (h) ∣ \leq ∥ μ - ν ∥_{T V}$ .

2.1step 1.2given

Apply step 1.2 with $μ = P^{n} (x, \cdot)$ and $ν = π$ . The given rate then yields $∣ P^{n} h (x) - π (h) ∣ \leq M (x) r^{n}$ for every $x$ .

3.1A1F1F2F3F5step 2.1given∎

Using [F5], [F2], and [F3], $∣ P (A \cap B) - P (A) P (B) ∣ = ∣ E [1_{A} (P^{n} h (X_{j}) - π (h))] ∣ \leq E ∣ P^{n} h (X_{j}) - π (h) ∣ \leq π (M) r^{n} .$ The first inequality uses $0 \leq 1_{A} \leq 1$ ; the last follows from the preceding pointwise bound and stationarity of $X_{j}$ . Since $A, B, j$ were arbitrary, taking the supremum in [F1] proves the claim.

Source notes

The proof is local: the Markov property converts separated future events to a bounded function of the state, and the total-variation bound controls its expectation. The source's factor convention is matched to the repository definition of total variation as a supremum over events.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

A measurable observation does not increase stationary mixing coefficients

Statement

Let $X = (X_{k})_{k \geq 0}$ be a strictly stationary process on a measurable state space and let $h$ be a measurable real-valued function. Put $Y_{k} = h (X_{k})$ . Then $Y$ is strictly stationary and, for every $n \geq 1$ , $α_{Y} (n) \leq α_{X} (n), ϕ_{Y} (n) \leq ϕ_{X} (n), ρ_{Y} (n) \leq ρ_{X} (n),$ where the coefficients are as in Strong, uniform, and maximal-correlation mixing coefficients.

Facts & Assumptions

Given: A strictly stationary process $X$ and a measurable real-valued map $h$ .

[F1]

Strict stationarity means every finite-dimensional distribution is invariant under a common nonnegative time shift. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

The coefficients are suprema over past and separated-future events, or over square-integrable variables measurable with respect to those sigma-algebras. (Strong, uniform, and maximal-correlation mixing coefficients)

Proof

1.1F1given

For any finite indices $k_{1}, \dots, k_{m}$ and time shift $t \geq 0$ , strict stationarity [F1] makes $(X_{k_{1} + t}, \dots, X_{k_{m} + t})$ equal in distribution to $(X_{k_{1}}, \dots, X_{k_{m}})$ . Applying the measurable map $h$ coordinatewise preserves equality in distribution, so $(Y_{k_{1} + t}, \dots, Y_{k_{m} + t})$ has the same law as $(Y_{k_{1}}, \dots, Y_{k_{m}})$ . Thus $Y$ is strictly stationary.

1.2F2given

For every $j$ , measurability gives $σ (Y_{0}, \dots, Y_{j}) \subseteq σ (X_{0}, \dots, X_{j})$ and $σ (Y_{j + n}, Y_{j + n + 1}, \dots) \subseteq σ (X_{j + n}, X_{j + n + 1}, \dots)$ . Therefore every event pair used in the suprema for $α_{Y}$ and $ϕ_{Y}$ is also allowed for $X$ , with the same probability difference. Taking the suprema in [F2] proves $α_{Y} (n) \leq α_{X} (n)$ and $ϕ_{Y} (n) \leq ϕ_{X} (n)$ .

2.1F2step 1.2∎

Every square-integrable past variable and future variable measurable with respect to the corresponding $Y$ -sigma-algebras is also measurable with respect to the corresponding $X$ -sigma-algebras. Its correlation is unchanged, so the supremum for $ρ_{Y} (n)$ is over a subset of the pairs allowed for $ρ_{X} (n)$ . Hence $ρ_{Y} (n) \leq ρ_{X} (n)$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Stationary Markov-chain total-variation convergence implies strong mixing

Statement

Assume AC. Let $(X, X)$ have a supplied countable algebra $C$ generating $X$ , and let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with invariant probability $π$ . If $∥ P^{n} (x, \cdot) - π ∥_{T V} ⟶ 0 for π -almost every x,$ then its stationary output process is strongly mixing: $α_{X} (n) ⟶ 0.$

Facts & Assumptions

Given: AC, the countable generating algebra $C$ , a stationary homogeneous $P$ -chain with invariant law $π$ , and the stated $π$ -almost-everywhere total-variation convergence.

[A1]

AC is the choice principle for the conditional classes used by the Markov-chain interfaces. (The Axiom of Choice)

[F1]

The strong-mixing coefficient is the supremum of $∣ P (A \cap B) - P (A) P (B) ∣$ over past and separated future events. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

The bounded-future-path Markov property gives a measurable function of the present state for a bounded future-path functional. (Markov property for bounded future path functionals)

[F3]

For bounded measurable $h$ , $E [h (X_{j + n}) ∣ F_{j}] = P^{n} h (X_{j})$ . (Chapman-Kolmogorov equations)

[F4]

Total variation is $∥ μ - ν ∥_{T V} = \sup_{A} ∣ μ (A) - ν (A) ∣$ . (Total variation distance for probability laws)

[F5]

A known measurable factor can be taken out of conditional expectation. (Taking out what is known)

[F6]

Increasing measurable sets have increasing measures converging to the measure of their union, and symmetric differences satisfy measure subadditivity. (Continuity from below for measures, Measure of a set difference when the smaller set has finite measure)

[F7]

Dominated convergence applies to measurable functions bounded by an integrable majorant. (Dominated convergence)

[F8]

Kernel integration makes $x \mapsto P^{n} h (x)$ measurable for bounded measurable $h$ , in particular $x \mapsto P^{n} (x, C)$ for measurable $C$ . (Measure kernel and probability kernel, Iterated transition kernels, Measurability of integration against a kernel)

Proof

1.1A1F2F3given

Fix $n \geq 1$ , $j \geq 0$ , $A \in F_{j}$ , and $B \in σ (X_{j + n}, X_{j + n + 1}, \dots)$ . By [F2], the indicator of this future-path event has conditional expectation $h (X_{j + n})$ for a measurable $0 \leq h \leq 1$ . Applying [F3] and the tower property gives $E [1_{B} ∣ F_{j}] = P^{n} h (X_{j})$ . Stationarity gives $P (B) = π (h)$ .

1.2F4construct

For probability measures $μ, ν$ and measurable $0 \leq h \leq 1$ , the dyadic simple approximation $h_{m} = 2^{- m} \sum_{r = 1}^{2^{m}} 1_{{h \geq r 2^{- m}}}$ differs from $h$ by at most $2^{- m}$ . Each indicator integral difference is bounded by [F4], so the integrals of $h_{m}$ differ by at most $∥ μ - ν ∥_{T V}$ . Letting $m \to \infty$ gives $∣ μ (h) - ν (h) ∣ \leq ∥ μ - ν ∥_{T V}$ .

1.3F4F6F8given

For fixed $n$ and $x$ , $μ_{n, x} : = P^{n} (x, \cdot) + π$ is a finite measure. Let $D$ be the sets approximable in $μ_{n, x}$ -symmetric-difference measure by members of $C$ . It contains $C$ and is closed under complements because $C$ is an algebra. For $A_{i} \in D$ and $ε > 0$ , [F6] gives $N \geq 1$ with $μ_{n, x} ((⋃_{i} A_{i}) ∖ (⋃_{i = 1}^{N} A_{i})) < ε / 2$ . Choose $C_{i} \in C$ with $μ_{n, x} (A_{i} △ C_{i}) < ε / (2 N)$ for $i \leq N$ . The finite union $C = ⋃_{i = 1}^{N} C_{i}$ belongs to $C$ and $μ_{n, x} ((⋃_{i} A_{i}) △ C) < ε$ by finite subadditivity. Thus $D$ is a sigma-algebra containing the generator $C$ , so $D = X$ . Approximating each measurable $A$ in this way shows $d_{n} (x) : = \sup_{C \in C} ∣ P^{n} (x, C) - π (C) ∣ = ∥ P^{n} (x, \cdot) - π ∥_{T V}$ : the discrepancy for $A$ differs from that for an approximating $C$ by at most $P^{n} (x, A △ C) + π (A △ C)$ . Since $C$ is countable and each evaluation is measurable by [F8], $d_{n}$ is measurable; also $0 \leq d_{n} \leq 1$ .

2.1F1F2F5F7step 1.1step 1.2step 1.3given∎

Using [F5] and step 1.1, $∣ P (A \cap B) - P (A) P (B) ∣ \leq E ∣ P^{n} h (X_{j}) - π (h) ∣ \leq \int d_{n} (x) π (d x)$ . The last inequality follows from step 1.2 applied pointwise to $P^{n} (x, \cdot)$ and $π$ , and then from step 1.3. By hypothesis $d_{n} (x) \to 0$ for $π$ -almost every $x$ ; since $d_{n} \leq 1$ , [F7] makes the integral tend to zero. This bound is independent of $j, A, B$ , so taking the supremum in [F1] proves $α_{X} (n) \to 0$ .

Source notes

The proof uses the supplied countable algebra to make the total-variation supremum a measurable function of the starting state. The bound on the strong-mixing coefficient then follows by dominated convergence.

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Uniform ergodicity bounds the uniform-mixing coefficient

Statement

Assume AC. Let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with invariant probability $π$ . If for constants $C < \infty$ and $r \in (0, 1)$ $\sup_{x \in X} ∥ P^{n} (x, \cdot) - π ∥_{T V} \leq C r^{n} (n \geq 1),$ then the uniform-mixing coefficient satisfies $ϕ_{X} (n) \leq C r^{n} (n \geq 1) .$

Facts & Assumptions

Given: AC, a stationary homogeneous Markov chain with kernel $P$ and invariant probability $π$ , and the displayed uniform rate.

[A1]

AC is the choice principle for the conditional classes used by the Markov-chain interfaces. (The Axiom of Choice)

[F1]

The uniform-mixing coefficient is the supremum of $∣ P (B ∣ A) - P (B) ∣$ over past events $A$ with positive probability and separated future events $B$ . (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

[F3]

For bounded measurable $h$ , $E [h (X_{j + n}) ∣ F_{j}] = P^{n} h (X_{j})$ . (Chapman-Kolmogorov equations)

[F4]

Total variation is $∥ μ - ν ∥_{T V} = \sup_{A} ∣ μ (A) - ν (A) ∣$ . (Total variation distance for probability laws)

[F5]

A known measurable factor can be taken out of conditional expectation. (Taking out what is known)

Proof

1.1A1F2F3F5given

Fix $n \geq 1$ , $j \geq 0$ , $A \in F_{j}$ with $P (A) > 0$ , and $B \in σ (X_{j + n}, X_{j + n + 1}, \dots)$ . By [F2], there is measurable $h : X \to [0, 1]$ with $E [1_{B} ∣ F_{j + n}] = h (X_{j + n})$ . By [F3] and [F5], $P (B ∣ A) = E [P^{n} h (X_{j}) ∣ A]$ . Stationarity gives $P (B) = π (h)$ .

1.2F4constructgiven

For probability measures $μ, ν$ and measurable $0 \leq h \leq 1$ , the dyadic simple functions $h_{m} = 2^{- m} \sum_{j = 1}^{2^{m}} 1_{{h \geq j 2^{- m}}}$ differ from $h$ by at most $2^{- m}$ . Their integrals differ by at most $∥ μ - ν ∥_{T V}$ , since each summand is an indicator and [F4] bounds its measure difference. Letting $m \to \infty$ gives $∣ μ (h) - ν (h) ∣ \leq ∥ μ - ν ∥_{T V}$ . Applying this with $μ = P^{n} (x, \cdot)$ and $ν = π$ proves $∣ P^{n} h (x) - π (h) ∣ \leq C r^{n}$ for every $x$ .

2.1F1step 1.1step 1.2∎

Averaging the pointwise bound of step 1.2 conditionally on $A$ in step 1.1 gives $∣ P (B ∣ A) - P (B) ∣ \leq C r^{n}$ . Taking the supremum in [F1] proves the result.

Source notes

Uniform total-variation convergence controls every conditional future event uniformly over the current state. No moment assumption on the observable is used in this mixing-coefficient bound.

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Zero covariance long-run variance gives a degenerate CLT under absolute summability

Statement

Let $(Y_{k})_{k \geq 0}$ be a centered, strictly stationary, square-integrable real process (The space $L^{p} (μ)$ as the quotient by null functions), and put $γ_{k} = Cov (Y_{0}, Y_{k})$ . If $\sum_{k \geq 1} ∣ γ_{k} ∣ < \infty$ and $γ_{0} + 2 \sum_{k = 1}^{\infty} γ_{k} = 0,$ then $n^{- 1 / 2} \sum_{k = 1}^{n} Y_{k} \to 0$ in $L^{2}$ and hence in distribution to $N (0, 0)$ .

Facts & Assumptions

Given: A centered strictly stationary real process with finite second moment and the stated covariance hypotheses.

[F1]

Variance is the centered second moment and covariance is the centered product expectation. (Moments, variance, and covariance on a probability space)

[F2]

Strict stationarity makes every covariance depend only on the lag. (Strong, uniform, and maximal-correlation mixing coefficients)

[F3]

Convergence in $L^{2}$ implies convergence in probability. ( $L^{p}$ convergence implies convergence in probability)

[F4]

Convergence in probability to a constant implies convergence in distribution to its point-mass law. (Convergence in probability implies convergence in distribution)

[F5]

Dominated convergence applies to a series indexed by counting measure when its terms have a summable majorant. (Dominated convergence)

Proof

1.1F1F2F5given

Expanding the finite square of $S_{n} : = \sum_{k = 1}^{n} Y_{k}$ and using centering and [F2] gives $\frac{1}{n} Var (S_{n}) = γ_{0} + 2 \sum_{k = 1}^{n - 1} (1 - \frac{k}{n}) γ_{k} .$ For each fixed $k$ , the weight tends to $1$ and its absolute value is at most $1$ . The summable sequence $∣ γ_{k} ∣$ dominates the summands, so [F5] gives $n^{- 1} Var (S_{n}) \to γ_{0} + 2 \sum_{k \geq 1} γ_{k} = 0$ .

2.1F1F3F4step 1.1∎

Since $E [S_{n}] = 0$ , step 1.1 gives $E [(S_{n} / \sqrt{n})^{2}] = n^{- 1} Var (S_{n}) \to 0$ . This is convergence to zero in $L^{2}$ ; [F3] gives convergence in probability, and [F4] gives convergence in distribution to the point mass at zero, namely $N (0, 0)$ .

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Poisson-equation martingale decomposition and MCMC CLT

Statement

Assume AC. Let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with invariant probability $π$ , natural filtration $F_{k}$ , and ergodic canonical path shift. Let $f$ be real Borel with $π (f) = 0$ and $f \in L^{2} (π)$ (The space $L^{p} (μ)$ as the quotient by null functions). Suppose there is a measurable $g \in L^{2} (π)$ such that $f = g - P g$ $π$ -almost everywhere, where $P g (x) = \int g (y) P (x, d y)$ . Put $D_{k + 1} : = g (X_{k + 1}) - P g (X_{k})$ and $τ_{f}^{2} : = E [D_{1}^{2}]$ . Then for every $n \geq 1$ , $\sum_{k = 0}^{n - 1} f (X_{k}) = g (X_{0}) - g (X_{n}) + \sum_{k = 0}^{n - 1} D_{k + 1},$ and $\frac{1}{\sqrt{n}} \sum_{k = 0}^{n - 1} f (X_{k}) \Rightarrow N (0, τ_{f}^{2}),$ including the degenerate case $τ_{f}^{2} = 0$ . If additionally $\sum_{k \geq 1} ∣ {Cov}_{π} (f (X_{0}), f (X_{k})) ∣ < \infty$ , then $τ_{f}^{2} = π (f^{2}) + 2 \sum_{k = 1}^{\infty} {Cov}_{π} (f (X_{0}), f (X_{k})),$ so this CLT's variance equals the covariance long-run variance of Long-run variance for a stationary Markov chain.

Facts & Assumptions

Given: AC, a stationary Markov chain with invariant probability $π$ , natural filtration $F_{k}$ , ergodic canonical shift, centered real $f \in L^{2} (π)$ , and a solution $g \in L^{2} (π)$ of $f = g - P g$ $π$ -almost everywhere.

[A1]

AC is the choice principle for the conditional-expectation classes used below. (The Axiom of Choice)

[F1]

For $g \in L^{2} (π)$ , $D_{k + 1} = g (X_{k + 1}) - P g (X_{k})$ is square-integrable, has conditional mean zero, and conditional variance $P (g^{2}) (X_{k}) - (P g (X_{k}))^{2}$ . (Kernel conditional expectations for square-integrable Markov observables)

[F2]

For a stationary ergodic process $Y$ , its averages converge almost surely to $E [Y_{0}]$ . (Birkhoff limit for a stationary integrable process)

[F3]

A square-integrable martingale-difference array with terminal conditional variance clock converging to $1$ in probability and vanishing unconditional Lindeberg sums has terminal sum converging in distribution to $N (0, 1)$ . (Martingale central limit theorem)

[F4]

For square-integrable real random variables, variance is the second centered moment and covariance is the centered product expectation. (Moments, variance, and covariance on a probability space)

[F5]

When the covariance series is absolutely convergent and the CLT is established, the covariance long-run variance is the sum $π (f^{2}) + 2 \sum_{k \geq 1} {Cov}_{π} (f (X_{0}), f (X_{k}))$ . (Long-run variance for a stationary Markov chain)

[F6]

A known measurable factor can be taken out of conditional expectation when products are integrable; Hölder bounds products of $L^{2}$ variables by their $L^{2}$ norms. (Taking out what is known, Holder's inequality for integrals, including the endpoint cases)

[F7]

If $U_{n} \Rightarrow U$ and $V_{n} \to 0$ in probability, then $U_{n} + V_{n} \Rightarrow U$ . (Converging together lemma)

[F8]

Dominated convergence passes limits through the integrals used below. (Dominated convergence)

[F9]

Convergence to zero in $L^{2}$ implies convergence to zero in probability. ( $L^{p}$ convergence implies convergence in probability)

[F10]

The canonical path law is shifted by the coordinate left shift; ergodic means the shift preserves the law and every invariant event has probability zero or one. (Canonical Markov chain on path space, Measure-preserving transformations and systems, Ergodicity relative to an invariant measure)

[F11]

A continuous scaling map sends weak convergence to the correspondingly scaled limiting law. (Continuous mapping theorem)

Proof

1.1A1F1given

By [F1], each $D_{k + 1}$ is a square-integrable martingale difference, so $E [D_{k + 1} ∣ F_{k}] = 0$ . Since $f = g - P g$ $π$ -almost everywhere and each $X_{k}$ has law $π$ , the identity holds at every $X_{k}$ simultaneously outside a countable union of null events. Therefore $f (X_{k}) = g (X_{k}) - g (X_{k + 1}) + D_{k + 1}$ almost surely.

1.2F1F2F10given

Set $q (x) = P (g^{2}) (x) - (P g (x))^{2}$ on the $π$ -full set where these terms are finite, and set $q = 0$ elsewhere. By [F1], $q (X_{k})$ is the conditional variance of $D_{k + 1}$ , is nonnegative, and $π (q) = E [D_{1}^{2}] = τ_{f}^{2}$ . Thus $q \in L^{1} (π)$ . The process $(q (X_{k}))_{k \geq 0}$ is a measurable factor of the stationary ergodic path process $(X_{k})$ : the inverse image of any shift-invariant event for $q (X)$ is shift-invariant for $X$ . Hence it too is stationary and ergodic by [F10], and [F2] gives $n^{- 1} \sum_{k = 0}^{n - 1} q (X_{k}) \to τ_{f}^{2}$ almost surely.

1.3F9givenalgebra

Stationarity and $g \in L^{2} (π)$ give $E [(g (X_{0}) - g (X_{n}))^{2} / n] \leq 4 π (g^{2}) / n \to 0$ . The $L^{2}$ -norm of the endpoint divided by $\sqrt{n}$ therefore tends to zero, so [F9] gives convergence to zero in probability.

2.1step 1.1algebra

Summing step 1.1 for $k = 0, \dots, n - 1$ cancels the intermediate $g (X_{k})$ terms and yields $S_{n} : = \sum_{k = 0}^{n - 1} f (X_{k}) = g (X_{0}) - g (X_{n}) + M_{n}$ , where $M_{n} : = \sum_{k = 0}^{n - 1} D_{k + 1}$ .

2.2A1F1F3F8F11step 1.1step 1.2givenconstruct

Suppose $τ_{f}^{2} > 0$ . For row $n$ , set $Z_{n, j} = D_{j} / \sqrt{n τ_{f}^{2}}$ for $1 \leq j \leq n$ , set $Z_{n, j} = 0$ for $j > n$ , and use $F_{n, j} = F_{\min (j, n)}$ for every $j \geq 0$ . These are square-integrable martingale-difference rows by step 1.1. Their terminal variance clock is $Γ_{n, \infty} = (n τ_{f}^{2})^{- 1} \sum_{k = 0}^{n - 1} q (X_{k}) \to 1$ almost surely by step 1.2. For every $ε > 0$ , stationarity of the transition pairs $(X_{j - 1}, X_{j})$ gives $\sum_{j = 1}^{n} E [Z_{n, j}^{2} 1_{{∣ Z_{n, j} ∣ > ε}}] = τ_{f}^{- 2} E [D_{1}^{2} 1_{{∣ D_{1} ∣ > ε \sqrt{n τ_{f}^{2}}}}] \to 0$ by [F8], since $D_{1}^{2}$ is integrable. Extending each row by zeros makes it an infinite array with the same terminal sum and clock. Therefore [F3] gives $M_{n} / \sqrt{n τ_{f}^{2}} \Rightarrow N (0, 1)$ ; scaling by $τ_{f}$ using [F11] yields $M_{n} / \sqrt{n} \Rightarrow N (0, τ_{f}^{2})$ .

3.1F1F7step 1.3step 2.1given

If $τ_{f}^{2} = 0$ , stationarity and [F1] imply $E [D_{k + 1}^{2}] = 0$ , so $D_{k + 1} = 0$ almost surely and hence $M_{n} = 0$ . By step 2.1, $S_{n} = g (X_{0}) - g (X_{n})$ . Step 1.3 shows $S_{n} / \sqrt{n} \to 0$ in probability; [F7] then gives convergence to the point mass $N (0, 0)$ .

3.2F7step 1.3step 2.1step 2.2algebra

If $τ_{f}^{2} > 0$ , step 2.2 gives $M_{n} / \sqrt{n} \Rightarrow N (0, τ_{f}^{2})$ . Combining this with step 1.3, the decomposition in step 2.1, and [F7] gives $S_{n} / \sqrt{n} \Rightarrow N (0, τ_{f}^{2})$ .

4.1F1F4F5F6F8step 2.1step 3.1step 3.2algebragiven∎

Assume additionally that the covariance series is absolutely convergent. Stationarity and expansion of the finite square give $n^{- 1} Var (S_{n}) = γ_{0} + 2 \sum_{k = 1}^{n - 1} (1 - k / n) γ_{k} \to γ_{0} + 2 \sum_{k = 1}^{\infty} γ_{k}$ , where $γ_{k} = {Cov}_{π} (f (X_{0}), f (X_{k}))$ ; [F8] applies to the series because $∣ γ_{k} ∣$ is summable. On the other hand, step 2.1 writes $S_{n} = R_{n} + M_{n}$ with $R_{n} = g (X_{0}) - g (X_{n})$ . For $i < j$ , $D_{i + 1}$ is $F_{j}$ -measurable and $E [D_{j + 1} ∣ F_{j}] = 0$ , so [F1, F6] give $E [D_{i + 1} D_{j + 1}] = 0$ . Stationarity gives $Var (M_{n}) = \sum_{j = 0}^{n - 1} E [D_{j + 1}^{2}] = n τ_{f}^{2}$ . Also $Var (R_{n}) \leq 4 π (g^{2})$ and [F6] gives $∣ Cov (R_{n}, M_{n}) ∣ \leq \sqrt{Var (R_{n}) Var (M_{n})} = O (\sqrt{n})$ . Dividing $Var (S_{n}) = Var (R_{n}) + Var (M_{n}) + 2 Cov (R_{n}, M_{n})$ by $n$ therefore shows that its limit is $τ_{f}^{2}$ . Steps 3.1 and 3.2 give the CLT, so [F5] identifies its variance with the displayed covariance sum.

Source notes

The decomposition is proved directly by telescoping. The CLT uses the conditional-variance ergodic theorem and the stated martingale-array CLT; finite variance of $f$ alone is not used as a CLT criterion. The covariance formula is proved only in the stated Poisson-equation case.

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Geometric small-set renewal convergence gives an integrable exponential hitting moment

Statement

Assume the Axiom of Choice. Let $Q$ be a probability kernel on any measurable space $(X, G)$ , with invariant probability $π$ . Let $C \in G$ satisfy $π (C) > 0$ , and suppose $Q (x, \cdot) \geq ϵ ν (\cdot) (x \in C), 0 < ϵ < 1,$ where $ν$ is a probability measure. Suppose some $A < \infty$ and $0 < θ < 1$ satisfy $∣ ν Q^{n} (C) - π (C) ∣ \leq A θ^{n} (n \geq 0) .$ For a $Q$ -chain started at $x$ , put $σ_{C} = \inf {n \geq 0 : X_{n} \in C}$ . Then some $s > 1$ makes $H (x) = E_{x} [s^{σ_{C}}] measurable, π (H) < \infty,$ where $s^{\infty} = \infty$ . In particular $H$ is finite $π$ -almost everywhere; finiteness at every state is not asserted.

For the same hypotheses, the split construction also has the following return and occupation properties. Put $w_{1} (x) = ϵ 1_{C} (x)$ and $w_{0} (x) = 1 - w_{1} (x)$ . Define $K_{x, 1} = ν$ , $K_{x, 0} = (Q (x, \cdot) - ϵ ν) / (1 - ϵ)$ on $C$ , and $K_{x, 0} = Q (x, \cdot)$ off $C$ . On $X \times {0, 1}$ use the kernel $\tilde{Q} ((x, i), d y \times {j}) = K_{x, i} (d y) w_{j} (y)$ and invariant law $\tilde{π} (d x, i) = w_{i} (x) π (d x)$ . Its atom $α = C \times {1}$ has mass $p = ϵ π (C) > 0$ . If $Z_{0}$ has law $κ = \tilde{π} (\cdot ∣ α)$ and $τ = \inf {n \geq 1 : Z_{n} \in α}$ , then some $b > 1$ satisfies $E_{κ} b^{τ} < \infty,$ and for every nonnegative measurable $g$ on the split space, $\tilde{π} (g) = p E_{κ} \sum_{k = 0}^{τ - 1} g (Z_{k}) .$ In particular $τ < \infty$ almost surely and $E_{κ} τ = 1 / p$ . These additional assertions use exactly the construction and hypotheses above; they do not assert an exponential entrance-time moment from every state.

Facts & Assumptions

Given: AC, the invariant law, small-set minorization, and scalar geometric-convergence bound in the Statement.

[A1]

The Axiom of Choice: AC is assumed, as required by the path-space and Markov suppliers below.

[F1]

Small-set minorization and geometric drift: the displayed minorization permits the probability residual $(Q (x, \cdot) - ϵ ν) / (1 - ϵ)$ on $C$ .

[F2]

Canonical Markov chain on path space: probability kernels on arbitrary measurable spaces have canonical path laws under AC.

[F3]

Measurability of integration against a kernel: integration of a nonnegative measurable function against a probability kernel is measurable.

[F4]

Discrete strong Markov property: future bounded path functionals obey the Markov identity at stopping times.

[F5]

Markov property for bounded future path functionals: bounded future-path expectations are measurable functions of the starting state and satisfy the deterministic-time Markov identity.

[F6]

Invariant initial law makes a Markov chain stationary: an invariant initial distribution makes every finite-dimensional law stationary.

[F7]

Monotone convergence for the integral: nonnegative increasing limits pass through integrals.

[F8]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative double sums and probability integrals may be interchanged.

[F9]

The sum of a complex power series is analytic throughout its open disc of convergence and Every complex analytic function is holomorphic: an absolutely convergent complex power series is holomorphic in its open disk of convergence.

[F10]

A complex power-series sum has complex derivatives of every order, obtained by repeated termwise differentiation: power series can be differentiated repeatedly term by term inside their convergence disk.

[F11]

All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle: a holomorphic function has derivatives of every order locally; these derivatives are continuous because each has a further complex derivative.

[F12]

Cauchy's inequalities bound the Taylor coefficients by the circle supremum: if $∣ g ∣ \leq B$ on a radius- $δ$ circle about $a$ contained with its disk in the holomorphic domain, then $∣ g^{(k)} (a) ∣ / k! \leq B δ^{- k}$ .

[F13]

Invariant and stationary distribution for a Markov kernel: invariance is equality of the initial law with its one-step image.

[F14]

Measure kernel and probability kernel: a probability kernel is measurable in its starting point and countably additive in its target event.

Proof

Given: The hypotheses in the Statement, with AC [A1].

1.1A1F1F2F3F14given

Set $w_{1} (x) = ϵ 1_{C} (x)$ and $w_{0} (x) = 1 - w_{1} (x)$ . On $\tilde{X} = X \times {0, 1}$ define $K_{x, 1} = ν$ ; define $K_{x, 0} = (Q (x, \cdot) - ϵ ν) / (1 - ϵ)$ for $x \in C$ , and $K_{x, 0} = Q (x, \cdot)$ otherwise. The split kernel is $\tilde{Q} ((x, i), B \times {j}) = \int_{B} w_{j} (y) K_{x, i} (d y) .$ Nonnegativity follows from [F1], measurability from [F3], and finite summation over the two flag sections defines a probability on every product-measurable event. Thus [F2] realizes the split path. If the initial flag at $x$ has probabilities $w_{i} (x)$ , summing it gives $\sum_{i} w_{i} (x) K_{x, i} = Q (x, \cdot)$ . The same summation at each subsequent time shows that the $x$ -coordinate path has the original $Q$ -chain law. The set $α = C \times {1}$ is an atom: all its states have the identical outgoing transition obtained by using $ν$ .

2.1F6F13step 1.1given

Put $\tilde{π} (d x, i) = w_{i} (x) π (d x)$ . Summing its incoming transitions first over $i$ gives $π Q = π$ and then assigns the new flag according to $w_{j}$ , proving $\tilde{π} \tilde{Q} = \tilde{π}$ by [F13]. Let $p = \tilde{π} (α) = ϵ π (C) > 0$ , and let $κ = \tilde{π} (\cdot ∣ α)$ , an ordinary normalized restriction. Write $P_{α} = P_{κ}$ ; its future after time zero is the same from every atom state. Starting in $α$ , the next coordinate law is $ν$ and subsequent coordinate transitions are $Q$ , so $u_{0} = 1, u_{n} = P_{α} (Z_{n} \in α) = ϵ ν Q^{n - 1} (C) (n \geq 1) .$ The hypothesis implies $∣ u_{n} - p ∣ \leq c θ^{n}$ for all $n \geq 0$ after increasing a finite $c$ .

3.1F4F7F8F9step 2.1

Let $τ = \inf {n \geq 1 : Z_{n} \in α}$ and $f_{n} = P_{α} (τ = n)$ for $n \geq 1$ , with $f_{0} = 0$ . Decompose a visit at time $n \geq 1$ by the first return time and use [F4] and the common atom transition to obtain $u_{n} = \sum_{j = 1}^{n} f_{j} u_{n - j}$ . The possible event $τ = \infty$ contributes zero. For $∣ z ∣ < 1$ , absolute convergence and [F8] give $U (z) = \sum_{n \geq 0} u_{n} z^{n} = \frac{1}{1 - F (z)}, F (z) = \sum_{n \geq 1} f_{n} z^{n} .$ Here $∣ F (z) ∣ \leq ∣ z ∣ \sum_{n} f_{n} \leq ∣ z ∣ < 1$ , so the denominator is legitimate. Since $u_{n} - p$ decays geometrically, [F9] gives $U (z) = p / (1 - z) + J (z)$ with $J$ holomorphic on $∣ z ∣ < θ^{- 1}$ . For real $t ↑ 1$ , $U (t) \to \infty$ ; hence $F (t) \to 1$ and [F7] gives $\sum_{n} f_{n} = 1$ .

3.2F6step 2.1algebra

Let $T = \inf {n \geq 0 : Z_{n} \in α}$ and start the split chain from $\tilde{π}$ . By [F6], $I_{j} = 1_{α} (Z_{j})$ have mean $p$ and $Cov (I_{0}, I_{n}) = p (u_{n} - p)$ . Expanding the square of $N^{- 1} \sum_{j = 0}^{N - 1} I_{j} - p$ and summing the geometric covariance bound from step 2.1 gives expectation at most $c_{1} / N$ for some finite $c_{1}$ . On ${T \geq N}$ this square equals $p^{2}$ , so $P_{\tilde{π}} (T \geq N) \leq c_{1} / (N p^{2}) \to 0$ . In particular the stationary split chain hits the atom almost surely; no unproved Harris or skeleton-recurrence theorem is needed.

4.1F7F10F11F12step 3.1

Near $z = 1$ the expression $\hat{F} (z) = 1 - \frac{1 - z}{p + (1 - z) J (z)}$ is holomorphic and agrees with $F$ on its overlap with $∣ z ∣ < 1$ . Indeed the denominator has value $p > 0$ at $1$ and remains nonzero on a sufficiently small disk; the quotient is complex differentiable there by the difference-quotient formula for the derivative of a reciprocal. By [F10], for $0 < t < 1$ and $k \geq 0$ , $F^{(k)} (t) = \sum_{n \geq k} f_{n} (n)_{k} t^{n - k}$ , where $(n)_{k} = n! / (n - k)!$ . Letting $t ↑ 1$ and using [F7] and derivative continuity from [F11] gives $\sum_{n \geq k} f_{n} (n)_{k} = {\hat{F}}^{(k)} (1)$ . Choose a radius $δ > 0$ whose closed disk about $1$ lies strictly within this holomorphic neighborhood and a finite bound $B$ for $∣ \hat{F} ∣$ on its circle. [F12] then bounds these nonnegative derivative values by $k! B δ^{- k}$ .

4.2F5F6F7F8step 2.1step 3.2

For bounded nonnegative $g$ , decompose $g (Z_{N})$ by the last atom visit at times $0, \dots, N$ . A last visit at time $N - k$ contributes $p E_{α} [g (Z_{k}); τ > k]$ , by stationarity and [F5]. The remaining paths have no atom visit through time $N$ , and their contribution $R_{N}$ lies between zero and $∥ g ∥_{\infty} P_{\tilde{π}} (T > N)$ . Thus $\tilde{π} (g) = p \sum_{k = 0}^{N} E_{α} [g (Z_{k}); τ > k] + R_{N} .$ Let $N \to \infty$ using step 3.2, then extend from bounded $g$ to all nonnegative measurable $g$ using [F7] and [F8]. This proves the stationary occupation identity $\tilde{π} (g) = p E_{α} \sum_{k = 0}^{τ - 1} g (Z_{k})$ . At $k = 0$ the atom state has law $κ$ ; subsequent transitions are independent of which atom state was chosen, so this argument does not require $g$ to be constant on $α$ .

5.1F8step 3.1step 4.1algebra

For $0 < h < δ$ , the finite binomial expansion for each integer $n$ , followed by [F8], yields $E_{α} [(1 + h)^{τ}] = \sum_{k \geq 0} \frac{h^{k}}{k!} \sum_{n \geq k} f_{n} (n)_{k} \leq \frac{B}{1 - h / δ} < \infty .$ Thus some $b > 1$ has $E_{α} b^{τ} < \infty$ . The return is almost surely finite by step 3.1. This argument establishes an exponential moment from positivity and analytic continuation; it does not assume in advance that the original series $F$ converges beyond the unit disk.

6.1F5F7F8step 1.1step 5.1step 4.2∎

Fix $1 < s < b$ and let $g (z) = E_{z} s^{T}$ , allowing infinity. The bounded path functions $s^{T \land l}$ increase to $s^{T}$ , so [F5] and [F7] give measurability and the Markov identity for this nonnegative function by truncation. On $α$ , $g = 1$ . For $k \geq 1$ , on ${τ > k}$ , the residual time to $α$ is $τ - k$ ; the Markov identity and step 4.2 give $\tilde{π} (g) = p (1 + E_{α} \sum_{k = 1}^{τ - 1} s^{τ - k}) \leq p E_{α} [τ s^{τ}] < \infty .$ The last moment is finite because $\sup_{j \geq 1} j (s / b)^{j} < \infty$ and step 5.1 bounds $E_{α} b^{τ}$ . For the coordinate chain initialized with the prescribed flag, its first $C$ visit $σ_{C}$ is no later than its first atom visit $T$ . Consequently $H (x) = E_{x} s^{σ_{C}}$ is measurable by the same bounded-path approximation and $π (H) \leq \tilde{π} (g) < \infty$ . This proves the conclusion, including the null-set qualification.

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Strong-mixing covariance bound in terms of tail quantiles

Statement

Let $A, B$ be sub-sigma-algebras of a probability space and put $a = \sup_{A \in A, B \in B} ∣ P (A \cap B) - P (A) P (B) ∣ .$ For any real random variable $U$ , define its absolute tail quantile by $Q_{U} (u) = \inf {t \geq 0 : P (∣ U ∣ > t) \leq u}, 0 < u < 1.$ This is a nonnegative nonincreasing Borel function and, for every nonnegative Borel function $ψ$ , $\int_{0}^{1} ψ (Q_{U} (u)) d u = E ψ (∣ U ∣) .$ If $U, V$ are square-integrable real random variables measurable respectively with respect to $A, B$ , then $∣ Cov (U, V) ∣ \leq 4 \int_{0}^{a} Q_{U} (u) Q_{V} (u) d u .$ In particular, if $∣ U ∣ \leq A$ and $∣ V ∣ \leq B$ almost surely, the right-hand side is at most $4 A B a$ . For complex $U, V$ with these bounds and the same measurability, $∣ E (U V) - E U E V ∣ \leq 16 A B a .$ For two collections of observations separated by $n$ time steps in a process $Y$ , the coefficient $a$ of their sigma-algebras is at most $α_{Y} (n)$ as defined in Strong, uniform, and maximal-correlation mixing coefficients. An integral whose upper endpoint is zero is zero.

Facts & Assumptions

Given: The sigma-algebras, quantiles, moment and measurability hypotheses in the Statement.

[F1]

The strong-mixing coefficient is the supremum of the event-probability differences for the specified separated sigma-algebras. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

Covariance is $E (U V) - E U E V$ for square-integrable real variables. (Moments, variance, and covariance on a probability space)

[F3]

Cauchy–Schwarz bounds the expectation of the absolute product of two square-integrable real variables. (Cauchy-Schwarz for random variables)

[F4]

A nonnegative random variable is the integral of its threshold indicators, and its expectation is its integrated tail probability. (Layer-cake formulas for random variables)

[F5]

Nonnegative product-measurable functions on sigma-finite product spaces admit either order of integration. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product)

[F6]

Finite measures agreeing on a generating pi-system and on the whole space are equal. (Finite measures agreeing on a generating pi-system and on the whole space are equal)

Proof

1.1F6givenalgebra

For a finite real $U$ , write $T (t) = P (∣ U ∣ > t)$ for $t \geq 0$ . The tail tends to zero at infinity, is nonincreasing and is right-continuous. Thus $Q_{U} (u)$ is finite for $u > 0$ , is nonincreasing, and is Borel measurable. More precisely, $Q_{U} (u) > t$ if and only if $u < T (t)$ : if $T (t) \leq u$ , then $t$ belongs to the set defining the infimum; if $T (t) > u$ , right-continuity supplies an $ε > 0$ with $T (t + ε) > u$ , so that infimum exceeds $t$ . Consequently Lebesgue measure on $(0, 1)$ gives $λ {u : Q_{U} (u) > t} = T (t)$ for every $t \geq 0$ . The laws of $Q_{U}$ under $λ$ and of $∣ U ∣$ therefore agree on all closed left rays and both have total mass one. These rays form a pi-system and generate the Borel sigma-algebra: their differences give half-open intervals, and countable unions of those give all open intervals. By [F6] the laws agree, proving the stated integral identity. In particular $\int_{0}^{1} Q_{U}^{2} = E U^{2}$ .

1.2F2F3F4F5given

Suppose first $U, V \geq 0$ and $U, V \in L^{2}$ . Their first moments and $E (U V)$ are finite by [F3]. The threshold-indicator identities and [F5] give $E (U V) = \int_{0}^{\infty} \int_{0}^{\infty} P (U > s, V > t) d s d t$ and $E U E V = \int_{0}^{\infty} \int_{0}^{\infty} P (U > s) P (V > t) d s d t$ . Both integrands have finite integrals, so their difference represents the covariance and the integral triangle inequality applies. The absolute threshold-indicator covariance is bounded by $a$ , by $P (U > s)$ , and by $P (V > t)$ , since the joint probability and the product probability both lie between zero and either marginal probability.

2.1F5step 1.1step 1.2algebra

Put $T_{U} (s) = P (U > s)$ and $T_{V} (t) = P (V > t)$ . By [F5] and step 1.1, $\int_{0}^{\infty} \int_{0}^{\infty} \min {a, T_{U} (s), T_{V} (t)} d s d t = \int_{0}^{a} Q_{U} (u) Q_{V} (u) d u .$ Indeed the minimum is $\int_{0}^{a} 1_{{u < T_{U} (s)}} 1_{{u < T_{V} (t)}} d u$ , and each inner threshold set has length its quantile. The right-hand side is finite by Cauchy–Schwarz on $(0, 1)$ and the squared-quantile identity in step 1.1; this particular integral Cauchy–Schwarz inequality follows by integrating $(Q_{U} - c Q_{V})^{2} \geq 0$ and minimizing the resulting quadratic, with the zero-norm case immediate. Step 1.2 therefore proves the covariance bound with constant one for nonnegative $U, V$ .

3.1F2step 2.1algebra

For real $U, V$ , expand $Cov (U^{+} - U^{-}, V^{+} - V^{-})$ into four terms. Each nonnegative part has a tail quantile at most the corresponding absolute quantile, because its tail probabilities are smaller. The result of step 2.1 bounds each term by $\int_{0}^{a} Q_{U} Q_{V}$ ; the triangle inequality gives the asserted factor four. If $∣ U ∣ \leq A$ and $∣ V ∣ \leq B$ , their absolute quantiles are bounded by $A, B$ , so the bound becomes $4 A B a$ , including $a = 0$ or either zero bound.

4.1F1step 3.1algebra∎

For complex $U, V$ , expand $E (U V) - E U E V$ into the four real bilinear covariance terms of their real and imaginary parts. Each real part is bounded in absolute value by $A$ or $B$ , so step 3.1 and the complex triangle inequality give $16 A B a$ . Finally, for observations separated by $n$ , both collection sigma-algebras are sub-sigma-algebras of the past and future fields appearing in [F1]; their event supremum is therefore at most $α_{Y} (n)$ . All assertions follow.

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Two-sided stationary extension preserves maximal-correlation bounds

Statement

Assume AC. Let $(Y_{k})_{k \geq 0}$ be a strictly stationary real sequence, with one-sided coefficients $ρ_{Y} (n)$ from Strong, uniform, and maximal-correlation mixing coefficients. There is a shift-invariant probability $Q$ on the coordinate sigma-algebra of $R^{Z}$ such that its coordinate process $(X_{k})_{k \in Z}$ restricted to $k \geq 0$ has the law of $Y$ . The invertible left shift $T (x)_{i} = x_{i + 1}$ preserves $Q$ . Put $F_{k} = σ (X_{i} : i \leq k)$ and $G_{k} = σ (X_{i} : i \geq k)$ . For every integer $k$ , every $n \geq 1$ , and real square-integrable $U$ measurable in $F_{k}$ and $V$ measurable in $G_{k + n}$ , $∣ {Cov}_{Q} (U, V) ∣ \leq ρ_{Y} (n) \sqrt{{Var}_{Q} (U) {Var}_{Q} (V)} .$ If $ρ_{Y} (n) \to 0$ , then $T$ is ergodic: every event $A$ with $Q (A △ T^{- 1} A) = 0$ has $Q (A) \in {0, 1}$ .

Facts & Assumptions

Given: AC and the strictly stationary real sequence $Y$ in the statement.

[A1]

AC is the inherited choice assumption. (The Axiom of Choice)

[F1]

Consistent finite-dimensional probability laws on standard-Borel coordinate spaces have a unique probability extension on the coordinate sigma-algebra, assuming AC. (Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces)

[F2]

Strict stationarity and $ρ_{Y} (n)$ use the finite nonnegative past and the entire future at each split. (Strong, uniform, and maximal-correlation mixing coefficients)

[F3]

Square-integrable products satisfy $E ∣ U V ∣ \leq ∥ U ∥_{2} ∥ V ∥_{2}$ . (Cauchy-Schwarz for random variables)

[F4]

Simple functions are dense in $L^{2}$ on a probability space. (Simple functions with finite-measure support are dense in $L^{p} (μ)$ for $1 \leq p < \infty$ )

[F5]

Two finite measures agreeing on a generating pi-system agree on its generated sigma-algebra. (Dynkin's pi-lambda theorem)

[F6]

Ergodicity requires invariant measurable events to have probability zero or one. (Ergodicity relative to an invariant measure)

Proof

1.1A1F1F2F5construct

For finite $F \subset Z$ , choose an integer $m \geq 0$ with $F + m \subset N_{0}$ and prescribe the law of $(X_{i})_{i \in F}$ to be that of $(Y_{i + m})_{i \in F}$ . If $m^{'}$ is another permissible shift, stationarity compares the smaller shift with the larger one, so the law is independent of the choice. Restrictions are consistent because finitely many coordinate sets can be translated by one common shift. Every coordinate space is the standard Borel space $R$ , so [F1] gives $Q$ . Its nonnegative restriction equals the process law of $Y$ on cylinders and hence everywhere by [F5]. Translating finite sets does not change their prescribed laws, so $Q \circ T^{- 1} = Q$ by the same uniqueness argument. Both $T$ and its inverse are measurable because their coordinate maps are coordinates.

1.2F4construct

For any subset $I \subset Z$ , let $A_{I}$ be the algebra of events depending on finitely many coordinates in $I$ . Every event in $σ (X_{i} : i \in I)$ can be approximated in symmetric-difference probability by $A_{I}$ . Indeed, the approximable events are closed under complements. For a countable union, first replace that union by a finite subunion with arbitrarily small lost probability, using continuity from below, then approximate its finitely many constituent events and apply the union bound. Thus the approximable events form a sigma-algebra containing $A_{I}$ . By [F4], approximate an $L^{2}$ variable by simple functions, and approximate the finitely many indicator events of each simple function by $A_{I}$ events. For an indicator the squared $L^{2}$ error is its symmetric-difference probability; finite summation gives the required arbitrarily small $L^{2}$ error for a simple function. Consequently every such variable has finite-coordinate $L^{2}$ approximants.

2.1F2F3step 1.1step 1.2

First suppose $U$ and $V$ depend on finitely many coordinates in $(- \infty, k]$ and $[k + n, \infty)$ , respectively. Translate all coordinates by one integer $m \geq 0$ so that the used past coordinates and the split $k + m$ are nonnegative. Their joint distribution is an original-process distribution by step 1.1. The translated variables are measurable in $σ (Y_{0}, \dots, Y_{k + m})$ and $σ (Y_{k + m + n}, Y_{k + m + n + 1}, \dots)$ , so [F2] proves the asserted covariance inequality when both variances are positive. A variable of variance zero equals its mean almost surely and has zero covariance, so the inequality holds then too. For arbitrary $U, V$ , take step 1.2's finite-coordinate $L^{2}$ approximants. Their means, second moments, and covariances converge by [F3], so passage to the limit proves the same inequality.

3.1F6step 1.1step 1.2step 2.1given∎

Assume $ρ_{Y} (n) \to 0$ , and let $A$ satisfy $Q (A △ T^{- 1} A) = 0$ , with $p = Q (A)$ . Given $ε > 0$ , step 1.2 supplies a finite-coordinate event $C$ with $Q (A △ C) < ε$ ; enlarge its coordinate support to $[- m, m]$ for some $m \geq 0$ . For $n > 2 m$ , $C$ is in $F_{m}$ and $T^{- n} C$ is in $G_{n - m}$ . Step 2.1 applied to their indicators gives $∣ Q (C \cap T^{- n} C) - Q (C)^{2} ∣ \leq ρ_{Y} (n - 2 m) / 4$ , since every indicator variance is at most $1 / 4$ . Invariance modulo null sets and measure preservation imply $Q (A \cap T^{- n} A) = p$ . The intersection changes in probability by at most $2 ε$ upon replacing both copies of $A$ by copies of $C$ , and $∣ p^{2} - Q (C)^{2} ∣ \leq 2 ε$ . Hence $∣ p - p^{2} ∣ \leq 4 ε + ρ_{Y} (n - 2 m) / 4$ . Let $n \to \infty$ and then $ε ↓ 0$ to obtain $p = p^{2}$ . Therefore $p \in {0, 1}$ , proving [F6] and the stronger mod-null formulation claimed.

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Moderate deviation density bounds for Gaussian anchored sums

Statement

Assume AC and the hypotheses of Gaussian anchored bounded sums have uniform cumulant and Fourier bounds, with $1 \leq d \leq 4$ , fixed $γ > 0$ , $τ = C_{0} B$ , $κ = τ / \sqrt{s}$ , and logarithmic moment generating function $K$ . Let $p_{h}$ denote the density of $S$ under the real exponential tilt with parameter $h$ . Every such density exists, is continuously differentiable and strictly positive, and $\sup_{y} p_{h} (y) \leq (2 π γ s)^{- d / 2} (h \in R^{d}) .$ There are constants $c_{γ} > 0$ , $C_{γ} < \infty$ , independent of $d, s, B, m$ and the summand laws, with the following property. If $κ \leq c_{γ}$ and $τ ∣ h ∣ \leq 1 / 8$ , put $μ_{h} = E_{h} S$ and $D_{h} = {cov}_{h} (S)$ . Then $\frac{7}{8} s I_{d} \leq D_{h} \leq \frac{9}{8} s I_{d} .$ For $x \in R^{d}$ put $r = ∣ D_{h}^{- 1 / 2} x ∣$ and let $q_{D_{h}} (x) = (2 π)^{- d / 2} (\det D_{h})^{- 1 / 2} e^{- r^{2} / 2}$ . Whenever $κ r \leq c_{γ}$ , $∣ \log \frac{p_{h} (μ_{h} + x)}{q_{D_{h}} (x)} ∣ \leq C_{γ} κ (1 + r^{3}) .$ For $r \geq 7$ in the same region, ${\frac{d}{d t} \log p_{h} (μ_{h} + x + t x) ∣}_{t = 0} \leq - \frac{1}{2} r^{2} .$ These assertions apply separately to every row isometry image in the supplier, including every required coordinate prefix and normalized joint child sum and difference. Their tilt parameter and covariance are taken in that image's own coordinates.

Facts & Assumptions

Given: The hypotheses of the Statement. The inverse square root of a positive definite matrix is its symmetric positive inverse square root, and directional derivatives at a point always keep the chosen exponential tilt fixed.

[F1]

The bounded and Gaussian sum has an analytic cumulant with $∣ K_{u v v} (z) ∣ \leq τ s ∣ u ∣ ∣ v ∣^{2}$ on $τ ∣ z ∣ < 1$ . Under every real tilt its Gaussian component remains independent with covariance $G \geq γ s I_{d}$ , its characteristic function has modulus at most $e^{- γ s ∣ t ∣^{2} / 2}$ , and its Fourier tails, with or without one linear frequency factor, decay faster than any fixed power of $κ$ . The same holds for every row isometry image. Gaussian anchored bounded sums have uniform cumulant and Fourier bounds.

[F2]

Positive definite Gaussian densities have the displayed quadratic form and total mass one; their characteristic functions are the matching Gaussian exponentials. A positive-definite multivariate normal law has its Gaussian density, Characteristic function of a multivariate normal law.

[F3]

Fubini, Tonelli and dominated convergence license absolutely integrable interchanges and dominated differentiation. Fubini's theorem for L^1 functions on a sigma-finite product, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Dominated convergence.

[F4]

Real symmetric matrices have orthonormal eigenbases; linear changes of Lebesgue variables multiply by the absolute determinant. Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis, A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions.

[F5]

Cauchy–Schwarz and integration of derivatives along line segments give the norm and Lipschitz estimates below. Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz.

[F6]

A Euclidean Cauchy sequence converges. $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ .

[A1]

AC is assumed and inherited through the Gaussian and integration suppliers. The Axiom of Choice.

Proof

1.1F1F2F3F4

Under any real tilt, write $S = Y + Z_{h}$ , where the two terms are independent and $Z_{h}$ is Gaussian of covariance $G$ by [A1, F1]. Tonelli and translation of the Gaussian density show that $p_{h} (y) = E_{h} q_{G} (y - Y - E_{h} Z_{h})$ is a probability density. It is strictly positive because its integrand is strictly positive, and is bounded by $(2 π)^{- d / 2} (\det G)^{- 1 / 2} \leq (2 π γ s)^{- d / 2}$ . The Gaussian density and each of its first derivatives are bounded and continuous, so dominated convergence gives continuous first derivatives of this convolution. This proves the global assertions without a general vector inversion theorem.

1.2F1F3F5algebra

Exponential moment differentiation identifies $K^{'} (h)$ and $K^{''} (h)$ with the tilted mean and covariance, respectively. To justify the differentiation, the bounded summands permit every polynomial factor, and the Gaussian square completion in [F1] makes all its polynomial moments integrable even after a slightly stronger real exponential; dominated convergence applies. For real $u, v, w$ , polarization of the symmetric complex valued form $Q (v, w) = K_{u v w} (z)$ gives $∣ Q (v, w) ∣ \leq τ s ∣ u ∣ ∣ v ∣ ∣ w ∣$ : apply $4 Q (v, w) = Q (v + w, v + w) - Q (v - w, v - w)$ after replacing $v, w$ by $a v, a^{- 1} w$ , and minimize the resulting bound $τ s ∣ u ∣ (a^{2} ∣ v ∣^{2} + a^{- 2} ∣ w ∣^{2}) / 2$ over $a > 0$ . Integrating $K^{'''}$ on the real segment from zero to $h$ therefore gives $∥ K^{''} (h) - s I ∥ \leq τ s ∣ h ∣$ . Hence $7 s I / 8 \leq D_{h} \leq 9 s I / 8$ for $τ ∣ h ∣ \leq 1 / 8$ .

2.1step 1.1F1F2F3F4

We will also need its Fourier formula. Integrating the density of $N_{d} (0, G^{- 1})$ against the character $e^{- i t \cdot y}$ and using its characteristic function in [F2] gives $q_{G} (y) = (2 π)^{- d} \int e^{- i t \cdot y} e^{- t^{T} G t / 2} d t$ . The constants follow directly by inserting that density and multiplying by $(2 π)^{- d / 2} (\det G)^{- 1 / 2}$ . Insert this formula in step 1.1 and use absolute Fubini, dominated by the integrable Gaussian $e^{- t^{T} G t / 2}$ . Thus $p_{h} (y) = (2 π)^{- d} \int e^{- i t \cdot y} φ_{h} (t) d t$ , and its derivative in direction $v$ is the same integral with factor $- i (t \cdot v)$ . This differentiation is dominated because $∣ t \cdot v ∣ e^{- γ s ∣ t ∣^{2} / 2}$ is integrable by [F1]. Centering or making an invertible linear change of coordinates gives the identical formulas for those transformed densities.

2.2step 1.2F1F4algebra

Fix this $h$ and center its law. Its cumulant is $K_{h} (z) = K (h + z) - K (h) - z \cdot μ_{h}$ . On $2 τ ∣ z ∣ < 1$ the original analytic ball still contains $h + z$ , since $τ ∣ h + z ∣ < 5 / 8$ . Its third derivative is bounded by $τ s ∣ u ∣ ∣ v ∣^{2} \leq 2 τ ∣ u ∣ v^{T} D_{h} v$ . Whiten by $D_{h}^{- 1 / 2}$ , writing $W = D_{h}^{- 1 / 2} (S - μ_{h})$ under this tilt and $L (a) = K_{h} (D_{h}^{- 1 / 2} a)$ . Then $L (0) = 0$ , $L^{'} (0) = 0$ , $L^{''} (0) = I$ , and, with $λ = 2 τ / \sqrt{λ_{\min} (D_{h})} \leq 3 κ$ , it is analytic on $λ ∣ a ∣ < 1$ and satisfies $∣ L_{u v v} (a) ∣ \leq λ ∣ u ∣ ∣ v ∣^{2}$ . The centered Gaussian component of $W$ has covariance at least $(8 γ / 9) I$ . Its covariance remains unchanged under every further real tilt; consequently all such tilted characteristic functions have modulus at most $e^{- (4 γ / 9) ∣ t ∣^{2}}$ .

3.1step 2.2F5algebra

Polarization as in step 1.2 and integration along $t a$ , $0 \leq t \leq 1$ , give $∥ L^{''} (a) - I ∥ \leq λ ∣ a ∣$ , $∣ L^{'} (a) - a ∣ \leq λ ∣ a ∣^{2} / 2$ , and $∣ L (a) - ∣ a ∣^{2} / 2 ∣ \leq λ ∣ a ∣^{3} / 6$ for real $a$ in the ball. For real $t$ the same last estimate along the complex segment $i s t$ gives $L (i t) = - ∣ t ∣^{2} / 2 + R (t)$ with $∣ R (t) ∣ \leq λ ∣ t ∣^{3} / 6$ : this uses only the real direction $t$ in each derivative and the factors $i^{3}$ from the segment parameter.

4.1step 3.1F5F6construct

Put $b = D_{h}^{- 1 / 2} x$ , $r = ∣ b ∣$ . If $λ r \leq 1 / 16$ , the map $T (a) = b - (L^{'} (a) - a)$ sends the closed ball $∣ a ∣ \leq 2 r$ into itself by step 3.1, and its Lipschitz constant there is at most $2 λ r \leq 1 / 8$ . Starting at zero, its iterates are Cauchy since consecutive distances decrease geometrically; their limit exists by [F6], lies in the closed ball, and is a fixed point by continuity. Call it $a$ . It satisfies $L^{'} (a) = b$ , $∣ a ∣ \leq 2 r$ , and $∣ a - b ∣ \leq 2 λ r^{2}$ . For $r = 0$ take $a = 0$ . Thus no unproved global inverse assertion is needed for the saddlepoint.

5.1step 2.2step 3.1step 4.1choose

Tilt $W$ further by the fixed parameter $a$ . Its mean is $b$ and covariance $A = L^{''} (a)$ . Put $ϵ = λ ∣ a ∣ \leq 2 λ r$ ; step 3.1 gives $(1 - ϵ) I \leq A \leq (1 + ϵ) I$ . Its cumulant centered at $b$ has third derivative bounded by a fixed multiple of $λ ∣ u ∣ ∣ v ∣^{2}$ on a complex ball of radius $c / λ$ , because $a$ lies in the interior of the original analytic ball by step 4.1. Its characteristic modulus on all real frequencies is bounded by the same Gaussian $e^{- (4 γ / 9) ∣ t ∣^{2}}$ from step 2.2. In what follows reduce the final constant $c_{γ}$ so that $ϵ \leq 1 / 100$ and all these interior margins hold.

6.1step 5.1F2F5algebra

Let $ψ (t)$ be that last centered characteristic function and let $g_{A} = (2 π)^{- d / 2} (\det A)^{- 1 / 2}$ . For $∣ t ∣ \leq a_{0} / λ$ , with a sufficiently small fixed $a_{0} > 0$ , the integrated third derivative estimate gives $\log ψ (t) = - t^{T} A t / 2 + E (t)$ , $∣ E (t) ∣ \leq C λ ∣ t ∣^{3}$ , and $Re \log ψ (t) \leq - c ∣ t ∣^{2}$ . The elementary identity $e^{z} - e^{w} = (z - w) \int_{0}^{1} e^{(1 - u) w + u z} d u$ therefore gives $∣ ψ (t) - e^{- t^{T} A t / 2} ∣ \leq C λ ∣ t ∣^{3} e^{- c ∣ t ∣^{2}}$ there. Its integral, and its integral after multiplication by $∣ t \cdot v ∣$ , are bounded by $C λ$ and $C λ ∣ v ∣$ , respectively. To check the integrability explicitly, $∣ t ∣^{j} e^{- c ∣ t ∣^{2} / 2}$ is bounded for each fixed $j$ , by differentiation in $∣ t ∣$ , and the remaining Gaussian is integrable by [F2]. Constants are uniform over $d \leq 4$ and $A$ in the displayed neighborhood of $I$ .

7.1step 2.1step 6.1F1F2F3F4algebra

Outside the ball of step 6.1, both $ψ$ and the Gaussian comparator have unweighted integral $O_{γ} (λ^{2})$ and weighted integral $O_{γ} (λ^{2} ∣ v ∣)$ . Indeed their global bounds are $e^{- c_{γ} ∣ t ∣^{2}}$ ; extract $e^{- c_{γ} a_{0}^{2} / (2 λ^{2})}$ and integrate the remaining Gaussian, also with $∣ t \cdot v ∣$ . The extracted exponential is smaller than any prescribed fixed power of $λ$ because $z^{j} e^{- c z^{2}}$ is bounded. This is the same direct Gaussian tail argument as [F1], and uses no smoothness condition on a lattice law. Combining this with step 6.1 and the Fourier formula of step 2.1 gives, for the density $f_{a}$ of the further tilted $W$ , $∣ f_{a} (b) - g_{A} ∣ \leq C_{γ} λ, \sup_{y} ∣ \partial_{v} f_{a} (y) ∣ \leq (\sqrt{2 / π} g_{A} \sqrt{v^{T} A^{- 1} v} + C_{γ} λ ∣ v ∣) .$ The Gaussian integral in the derivative bound is exact: transform by $A^{1 / 2}$ and rotate $A^{- 1 / 2} v$ onto the first coordinate; the standard Gaussian first absolute moment is $\sqrt{2 / π}$ , from $\int_{0}^{\infty} t e^{- t^{2} / 2} d t = 1$ .

8.1step 5.1step 7.1F4algebrachoose

The eigenvalue bounds in step 5.1 give $∣ \log \det A ∣ \leq C d ϵ$ because $∣ \log (1 + u) ∣ \leq 2 ∣ u ∣$ for $∣ u ∣ \leq 1 / 2$ , by integrating $(1 + u)^{- 1}$ . Thus $g_{A}$ is bounded above and away from zero, uniformly, and $\log (g_{A} / (2 π)^{- d / 2}) = O (λ r)$ . Step 7.1 and $∣ \log (1 + u) ∣ \leq 2 ∣ u ∣$ imply $f_{a} (b) > 0$ and $\log \frac{f_{a} (b)}{(2 π)^{- d / 2}} = O_{γ} (λ (1 + r)) .$ We choose $c_{γ}$ small enough that all relative errors used here are at most $1 / 100$ .

9.1step 3.1step 4.1step 8.1F4algebra

The exact exponential tilt density identity is $f (y) = f_{a} (y) e^{L (a) - a \cdot y}$ , where $f$ is the original density of $W$ . It initially follows almost everywhere from the definition of the tilt; both sides are continuous by step 1.1, so it holds everywhere, since a continuous nonzero difference would be nonzero on a ball of positive Lebesgue measure. Step 3.1 and $∣ a - b ∣ \leq 2 λ r^{2}$ give $L (a) - a \cdot b = - r^{2} / 2 + O (λ r^{3}) .$ Indeed $∣ a ∣^{2} / 2 - a \cdot b = - ∣ b ∣^{2} / 2 + ∣ a - b ∣^{2} / 2$ , whose error is $O (λ^{2} r^{4}) = O (λ r^{3})$ because $λ r$ is small. Combining this identity with step 8.1 proves $∣ \log (f (b) / q_{I} (b)) ∣ \leq C_{γ} λ (1 + r^{3})$ . The constant Jacobian for whitening cancels in the density ratio, and $λ \leq 3 κ$ gives the stated moderate deviation bound.

10.1step 4.1step 5.1step 7.1step 8.1step 9.1algebrachoose

Differentiate the exact identity in step 9.1 in direction $b$ at $y = b$ , keeping $a$ fixed. It gives $\partial_{b} \log f (b) = (\partial_{b} f_{a} (b)) / f_{a} (b) - a \cdot b$ . By step 7.1, step 8.1, and $A = I + O (λ r)$ , reduce $c_{γ}$ further so that $∣ \partial_{b} f_{a} (b) ∣ / f_{a} (b) \leq 1.4 r$ ; this is possible because its limiting coefficient is $\sqrt{2 / π} < 1$ , the error is $O_{γ} (λ)$ , and $A$ tends uniformly to $I$ as $λ r \to 0$ . Also step 4.1 gives $a \cdot b \geq r^{2} - 2 λ r^{3} \geq .99 r^{2}$ after the same reduction. Hence $\partial_{b} \log f (b) \leq 1.4 r - .99 r^{2} \leq - r^{2} / 2$ for $r \geq 7$ . Whitening identifies this derivative with the derivative in direction $x$ in the Statement.

11.1step 2.2step 9.1step 10.1F1choose∎

There are only finitely many smallness requirements above: bounds on $κ$ , $λ r$ , $ϵ$ , and fixed multiples depending only on $γ$ of $λ$ . Since $λ \leq 3 κ$ , one sufficiently small positive $c_{γ}$ enforces all of them whenever $κ \leq c_{γ}$ and $κ r \leq c_{γ}$ . All constants are uniform over dimensions one through four. Apply the same argument to a row isometry image using [F1]; its Gaussian component retains the same covariance lower bound, and every step applies in its own coordinates. This proves the remaining assertions.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

A geometrically ergodic chain has an all-start standard-Borel observation factor

Statement

Assume AC. Let $P$ be a probability kernel on an arbitrary measurable space $(X, F)$ , with invariant probability $π$ , finite measurable $M : X \to [0, \infty)$ and $0 < r < 1$ satisfying $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) r^{n} (x \in X, n \geq 1) .$ Let $f : X \to R$ be finite measurable. For each specified initial law $μ$ , there are a nonempty standard-Borel space $S$ , a probability kernel $T$ on $S$ , an invariant probability $\hat{π}$ , an initial probability $\hat{μ}$ , and finite measurable functions $φ : S \to R$ and $\hat{M} : S \to [0, \infty)$ such that $∥ T^{n} (z, \cdot) - \hat{π} ∥_{T V} \leq \hat{M} (z) r^{n} (z \in S, n \geq 1) .$ The complete observation path $(f (X_{n}))_{n \geq 0}$ of a $μ$ -started $P$ -chain has the same law as $(φ (Y_{n}))_{n \geq 0}$ of a $\hat{μ}$ -started $T$ -chain. The corresponding stationary observation paths also have the same law, and $\hat{π} (∣ φ ∣^{a}) = π (∣ f ∣^{a})$ for every $a > 0$ , with infinity allowed. The factor may depend on $μ$ .

Moreover, any joint law of that observation path with a random element of a standard-Borel space can be lifted to an extension preserving the entire original chain probability space and process.

Facts & Assumptions

Given: AC and the objects and geometric bound in the Statement.

[A1]

The Axiom of Choice: AC supplies the inherited coding, factorization, disintegration and extension conventions.

[F1]

A kernel has a countably generated stable sub-sigma-field containing a given observable: a real observable lies in a countably generated kernel-stable sub-sigma-field, with restricted iterates and invariant law.

[F2]

Standard borel spaces admit bimeasurable real codings, Finite products of standard Borel spaces are standard Borel, and The real line with its Borel sigma-algebra is standard Borel: real finite tuples and standard-Borel spaces have Borel real codings.

[F3]

Doob-Dynkin factorization through the sigma-algebra generated by a function: a real or extended-real function measurable for the sigma-field of a real code factors through a Borel function of that code.

[F4]

Disintegration of a joint law on standard borel spaces: standard-Borel joint laws have measurable conditional probability kernels, unique outside a single null set.

[F5]

Dynkin's pi-lambda theorem: probability measures agreeing on a generating algebra agree everywhere.

[F6]

Markov property for bounded future path functionals: the original Markov conditional identities hold, and bounded future-observation expectations are measurable.

[F7]

Ionescu-Tulcea construction of a Markov chain: supplied measurable probability kernels extend an initial probability space while preserving its marginal.

[F8]

Monotone convergence for the integral and Measurability of integration against a kernel: nonnegative limits and probability-kernel integrals have their usual measurability and integral identities.

[F9]

The usual real line is Polish, as proved in The real line with its Borel sigma-algebra is standard Borel. Closed subsets of a metric space are $G_{δ}$ , so Under the Axiom of Countable Choice, countable products and $G_{δ}$ subspaces of Polish spaces are Polish makes the closed Cantor space and the countable product $R^{N_{0}}$ Polish under AC. Their Borel measurable spaces are standard Borel by Standard Borel spaces.

[F10]

Invariant and stationary distribution for a Markov kernel and Iterated transition kernels: invariance is $π P = π$ , and iterates are successive kernel integrations.

Proof

Given: The hypotheses in the Statement and AC [A1].

1.1A1F1F2F8F10constructgiven

Define $λ = \frac{1}{2} π + \frac{1}{2} \sum_{k \geq 0} 2^{- k - 1} μ P^{k} .$ This is a probability by [F8], $π, μ ≪ λ$ , and $λ P ≪ λ$ : if $λ (A) = 0$ , then $π (A) = 0$ and $μ P^{k} (A) = 0$ for every $k$ , which also makes $λ P (A) = 0$ . Thus $λ P^{j} ≪ λ$ for every $j$ . Apply [F2] to code the pair $(f, M)$ into one real observable, and apply [F1] to it. The resulting countably generated $G \subseteq F$ is kernel-stable and contains both $f$ and $M$ , since the coding has a Borel inverse.

2.1A1F3F9step 1.1construct

Enumerate generators $A_{j}$ of $G$ and set $c (x) = \sum_{j \geq 1} 2 \cdot 3^{- j} 1_{A_{j}} (x)$ , taking values in the closed Cantor subset $C$ of $[0, 1]$ . Its coordinate digits recover every $A_{j}$ , so $σ (c) = G$ . The target $C$ is standard Borel by [F9]; no Borel-image assertion about $c (X)$ is needed. By [F3], write $f = φ_{0} \circ c$ and $M = m_{0} \circ c$ for Borel extended-real functions on $C$ . Put $λ_{c} = c_{*} λ$ , $π_{c} = c_{*} π$ and $μ_{c} = c_{*} μ$ .

3.1A1F3F4F5F7F8F10step 1.1step 2.1

Push the probability $λ (d x) P (x, d y)$ to $(c (x), c (y)) \in C^{2}$ and disintegrate by [F4], obtaining a Borel kernel $T : C \to C$ . For each $B$ in a fixed countable generating algebra of $C$ , kernel stability and [F3] factor $P (x, c^{- 1} B)$ through $c (x)$ . Its value is the conditional mean of $1_{B} (c (X_{1}))$ given $c (X_{0})$ under this reference joint law. Hence $T (c (x), B) = P (x, c^{- 1} B)$ for $λ$ -almost every $x$ . Intersect over the countable algebra and apply [F5] to obtain equality of these probability measures simultaneously on every Borel $B$ , on one $λ$ -full set. In particular $λ_{c} T = c_{*} (λ P) ≪ λ_{c}$ , and $π_{c} T = π_{c}$ because $π ≪ λ$ and $π P = π$ .

4.1F8F10step 1.1step 3.1given

Iterating that kernel equality gives $T^{n} (c (x), \cdot) = c_{*} P^{n} (x, \cdot)$ for $λ$ -almost every $x$ , simultaneously for all $n \geq 0$ . At the induction step, $P (x, \cdot)$ avoids the preceding $λ$ -null exceptional set for $λ$ -almost every $x$ , since $λ P ≪ λ$ ; integration of a nonnegative Borel function of $c$ agrees under the two one-step measures by simple approximation and [F8]. The geometric hypothesis therefore implies $∥ T^{n} (z, \cdot) - π_{c} ∥_{T V} \leq m_{0} (z) r^{n}$ for $λ_{c}$ -almost every $z$ , simultaneously for $n \geq 1$ .

5.1F8F9step 2.1step 3.1step 4.1construct

On $C$ the total-variation supremum can be taken over its fixed countable generating algebra. Indeed, for any two probabilities, every Borel set is approximable in symmetric-difference measure by that algebra under their finite sum: the approximable sets form a sigma-field, using complements and first approximating a countable union by a finite union. Its probability difference is approximated too. Thus the supremum over the algebra equals the full supremum and is measurable as a function of the starting point. Let $B_{0}$ be the Borel set where all the bounds in step 4.1 hold and $φ_{0}, m_{0}$ are finite with $m_{0} \geq 0$ . It is $λ_{c}$ -full by step 2.1 and the finiteness of $f, M$ . Define $B_{j + 1} = B_{j} \cap {z : T (z, B_{j}) = 1}$ and $B = ⋂_{j \geq 0} B_{j}$ . Absolute continuity $λ_{c} T ≪ λ_{c}$ makes every $B_{j}$ full. If $z \in B$ , continuity from above gives $T (z, B) = 1$ . Thus $B$ is nonempty, Borel, absorbing and $λ_{c}$ -full; no Polish topology on $B$ is asserted.

6.1F8F9F10step 3.1step 5.1construct

Keep the factor space $S = C$ , and define $T_{*} (z, \cdot) = T (z, \cdot)$ for $z \in B$ and $T_{*} (z, \cdot) = π_{c}$ otherwise. This is a Borel probability kernel. Set $\hat{π} = π_{c}$ and $\hat{μ} = μ_{c}$ ; both assign mass one to $B$ , since they are absolutely continuous with respect to $λ_{c}$ . Define $φ = φ_{0}$ and $\hat{M} = m_{0}$ on $B$ , and set both to zero on $C ∖ B$ ; these functions are finite measurable. Since $B$ is absorbing and $π_{c} T = π_{c}$ , we have $π_{c} T_{*} = π_{c}$ . For $z \in B$ , the iterates of $T_{*}$ equal those of $T$ , so the geometric bound holds by step 5.1. For $z \notin B$ , $T_{*}^{n} (z, \cdot) = π_{c}$ for every $n \geq 1$ , so it holds with $\hat{M} (z) = 0$ . Thus $T_{*}$ is the required kernel on the nonempty standard-Borel space $C$ .

7.1F3F5F6F8step 1.1step 2.1step 3.1step 6.1

For an original chain with initial law $μ$ , every $X_{k}$ has law $μ P^{k} ≪ λ$ . Thus step 3.1 and [F6], followed by conditioning on the coded past, show that $c (X_{k})$ is a $T$ -chain. Since every coded state belongs to $B$ almost surely, this is also a $T_{*}$ -chain. Countably many time and algebra tests suffice; [F5] and bounded approximation extend them to every bounded Borel test. The code chain remains in $B$ almost surely and $f (X_{k}) = φ (c (X_{k}))$ simultaneously for every $k$ . The same argument applies with initial law $π$ . This proves the complete initial and stationary observation-path equalities and, by pushforward integration, every stated stationary moment identity.

8.1A1F4F7F8F9step 7.1∎

The real observation-path space $R^{N_{0}}$ is standard Borel: its product topology is Polish by [F9], and its Borel sigma-algebra is the coordinate product sigma-algebra, since the real line has a countable base and finite-coordinate basic open sets form a countable base for this product. Disintegrate any specified joint law of this path and a standard-Borel auxiliary random element conditional on the path by [F4]. Evaluate that kernel at the actual original observation path and extend the original probability space by its kernel integral as in [F7]; explicitly, $P (d ω) K ((f (X_{n} (ω)))_{n \geq 0}, d v)$ is countably additive by [F8] and has marginal $P$ . Its observation/auxiliary joint law is exactly the specified law. This proves the lifting assertion without a conditional kernel on the original arbitrary state space.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Geometric ergodicity and reversibility give exponential rho-mixing

Statement

Assume AC. Let $P$ be a probability kernel on $(X, X)$ with invariant probability $π$ , and consider its stationary output process $X$ . Assume it is $ϕ$ -irreducible (Phi-irreducibility for a general-state Markov kernel), aperiodic (Aperiodicity for a general-state Markov kernel), and Harris recurrent relative to its nonzero $σ$ -finite irreducibility measure $ϕ$ (Harris recurrence relative to an irreducibility measure). Suppose it is geometrically ergodic in the sense of Geometric and uniform ergodicity for MCMC CLTs and reversible with respect to $π$ in the symmetric-flow sense of General-state reversibility by symmetric flow measure. Then there are $C < \infty$ and $r \in (0, 1)$ such that its maximal-correlation mixing coefficient satisfies $ρ_{X} (n) \leq C r^{n} (n \geq 1) .$

Facts & Assumptions

Given: AC, the stationary $P$ -chain and all the hypotheses in the Statement.

[A1]

AC covers the conditional-expectation and path-law interfaces used below. (The Axiom of Choice)

[F1]

Geometric ergodicity supplies a measurable finite $M \geq 0$ and a common $r \in (0, 1)$ with $∥ P^{k} (x, \cdot) - π ∥_{T V} \leq M (x) r^{k}$ for every $x$ and $k \geq 1$ . (Geometric and uniform ergodicity for MCMC CLTs)

[F2]

Reversibility means that the probability flow measure $π (d x) P (x, d y)$ is invariant under coordinate interchange. (General-state reversibility by symmetric flow measure)

[F3]

The square-integrable kernel conditional-expectation identity holds with zero filling on the marginal-null set where an integral is not finite. (Kernel conditional expectations for square-integrable Markov observables)

[F4]

For square-integrable real variables, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ . (Cauchy-Schwarz inequality for $L^{2}$ )

[F5]

Dominated convergence passes almost-everywhere limits through integrals under an integrable dominating function. (Dominated convergence)

[F6]

A bounded measurable future-path functional has conditional expectation, given the chain's past through its initial time, equal to a bounded measurable function of the state at that time. (Markov property for bounded future path functionals)

[F7]

For bounded measurable $g$ , $E [g (X_{j + n}) ∣ F_{j}] = P^{n} g (X_{j})$ , where $F_{j} = σ (X_{0}, \dots, X_{j})$ . (Chapman-Kolmogorov equations)

[F8]

Nested conditional expectations satisfy the tower property, and a known factor can be taken out when the products are integrable. (Tower property of conditional expectation, Taking out what is known)

[F9]

Conditional Jensen gives $∣ E [V ∣ G] ∣^{2} \leq E [V^{2} ∣ G]$ . (Conditional jensen inequality)

[F10]

The coefficient $ρ_{X} (n)$ is the supremum of absolute correlations over square-integrable past and separated future variables of positive variance. (Strong, uniform, and maximal-correlation mixing coefficients)

Proof

1.1A1F2F3F4F5F7F9given

On real $L^{2} (π)$ put $T f = P f$ and $H = {f : π (f) = 0}$ . By [F3], conditional Jensen and stationarity give $∥ T f ∥_{2}^{2} \leq π (f^{2})$ and $π (T f) = π (f)$ . These identities also show that the operator is well defined on almost-everywhere classes. Thus $T$ is a contraction preserving $H$ , and $T 1 = 1$ . For bounded $f, g$ , [F2] gives $⟨ f, T g ⟩ = ⟨ T f, g ⟩$ by integrating $f (x) g (y)$ against the flow measure. Bounded truncation converges in $L^{2}$ by [F5], so contraction and [F4] extend this symmetry to all $L^{2}$ functions. Iterated kernel integration agrees with $T^{k}$ ; for bounded functions this is [F7], and contraction extends the identity to $L^{2}$ classes.

1.2F1constructalgebra

For probability measures $μ, ν$ and $0 \leq a \leq 1$ , approximate $a$ uniformly by $2^{- q} \sum_{i = 1}^{2^{q}} 1_{{a \geq i 2^{- q}}}$ . Each indicator integral difference is bounded by the event-supremum TV distance, so $∣ μ (a) - ν (a) ∣ \leq ∥ μ - ν ∥_{T V}$ on taking limits. Applying this to $a = (h + ∥ h ∥_{\infty}) / (2 ∥ h ∥_{\infty})$ for nonzero bounded measurable $h$ , and separating $h = 0$ , gives $∣ P^{k} h (x) - π (h) ∣ \leq 2 ∥ h ∥_{\infty} M (x) r^{k}$ .

2.1F1step 1.1step 1.2algebra

Put $C_{m} = {M \leq m}$ for positive integers $m$ . If bounded $h$ vanishes off $C_{m}$ and $f = h - π (h)$ , symmetry and invariance give $∥ T^{k} f ∥_{2}^{2} = ⟨ f, T^{2 k} f ⟩ = \int h (x) (P^{2 k} h (x) - π (h)) π (d x)$ for $k \geq 1$ . Step 1.2 therefore gives $0 \leq ∥ T^{k} f ∥_{2}^{2} \leq 2 m ∥ h ∥_{\infty}^{2} r^{2 k}$ . Write $A = \sqrt{2 m} ∥ h ∥_{\infty} < \infty$ , so $∥ T^{k} f ∥_{2} \leq A r^{k}$ . No integral of $M$ over the whole state space has been used.

3.1F4step 1.1step 2.1algebra

For $u \in H$ and $a \geq 1$ , symmetry and [F4] give $∥ T^{a} u ∥_{2}^{2} = ⟨ u, T^{2 a} u ⟩ \leq ∥ u ∥_{2} ∥ T^{2 a} u ∥_{2}$ . Repeating with $a = 1, 2, 4, \dots, 2^{ℓ - 1}$ yields $∥ T u ∥_{2} \leq ∥ u ∥_{2}^{1 - 2^{- ℓ}} ∥ T^{2^{ℓ}} u ∥_{2}^{2^{- ℓ}}$ . For the $f$ in step 2.1, if $f = 0$ the bound $∥ T f ∥_{2} \leq r ∥ f ∥_{2}$ is immediate; otherwise $A > 0$ and the last display is at most $r A^{2^{- ℓ}} ∥ f ∥_{2}^{1 - 2^{- ℓ}}$ , tending to $r ∥ f ∥_{2}$ . Hence that bound holds for every such $f$ .

4.1F1F4F5step 1.1step 3.1construct

For $u \in H$ choose a finite measurable representative and put $h_{m} = 1_{C_{m}} \max (- m, \min (u, m))$ . Finiteness of $M$ gives $h_{m} \to u$ almost everywhere and $∣ h_{m} - u ∣ \leq ∣ u ∣$ , so [F5] gives $h_{m} \to u$ in $L^{2}$ . By [F4], $π (h_{m}) \to 0$ , and therefore $h_{m} - π (h_{m}) \to u$ in $L^{2}$ . Applying step 3.1 to these centered functions and taking limits through the contraction $T$ gives $∥ T u ∥_{2} \leq r ∥ u ∥_{2}$ . Iteration on $H$ proves $∥ T^{n} u ∥_{2} \leq r^{n} ∥ u ∥_{2}$ for all $n \geq 1$ . This also covers $H = {0}$ .

5.1A1F4F6F7F8F9step 4.1given

Fix $j \geq 0$ and bounded centered real $U, V$ measurable respectively for $F_{j}$ and $σ (X_{j + n}, X_{j + n + 1}, \dots)$ . By [F6] there is bounded measurable $g$ such that $E [V ∣ F_{j + n}] = g (X_{j + n})$ . Stationarity and [F9] give $π (g) = 0$ and $∥ g ∥_{2} \leq ∥ V ∥_{2}$ . The tower property and [F7] give $E [V ∣ F_{j}] = P^{n} g (X_{j})$ . Taking out $U$ , applying [F4] and stationarity, and then step 4.1, yields $∣ E [U V] ∣ = ∣ E [U P^{n} g (X_{j})] ∣ \leq ∥ U ∥_{2} ∥ T^{n} g ∥_{2} \leq r^{n} ∥ U ∥_{2} ∥ V ∥_{2}$ .

6.1F4F5F10step 5.1∎

For general centered square-integrable $U, V$ in the same sigma-algebras, truncate each at levels $\pm m$ and subtract its expectation. By [F5] and [F4], these bounded centered variables converge in $L^{2}$ ; [F4] passes their covariances and norms to the limit in step 5.1. Dividing by the positive standard deviations and taking the suprema in [F10] proves $ρ_{X} (n) \leq r^{n}$ . If no positive-variance variables exist, the coefficient is zero by the nonnegative empty-supremum convention. Thus the stated conclusion holds with $C = 1$ .

Source notes

Jones, Theorem 2(3), supplies the stated implication. Roberts and Rosenthal, Geometric ergodicity and hybrid Markov chains, Theorem 2 of the author-hosted preprint (published Theorem 2.1), proves the reversible spectral-gap route; its full proof is on PDF pp. 7–10 at https://www.probability.ca/jeff/ftpdir/hybrid.pdf . The proof here instead uses envelope level sets and repeated Cauchy–Schwarz to obtain the operator bound locally. It needs neither an integrable envelope nor a compact or general spectral theorem. The irreducibility, recurrence and aperiodicity hypotheses are retained from the source statement; the explicit common geometric rate and reversibility suffice for this argument.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

A rho-mixing CLT for square-integrable stationary sequences

Statement

Assume AC. Let $(Y_{k})_{k \geq 0}$ be a centered strictly stationary real sequence with $E [Y_{0}^{2}] < \infty$ . If $\sum_{n = 1}^{\infty} ρ_{Y} (n) < \infty,$ then the covariance series $σ_{Y}^{2} : = E [Y_{0}^{2}] + 2 \sum_{k = 1}^{\infty} Cov (Y_{0}, Y_{k})$ converges absolutely and is nonnegative. If $σ_{Y}^{2} > 0$ , then for $S_{n} = \sum_{k = 1}^{n} Y_{k}$ , $n^{- 1 / 2} S_{n} \Rightarrow N (0, σ_{Y}^{2}) .$ If $σ_{Y}^{2} = 0$ , Zero covariance long-run variance gives a degenerate CLT under absolute summability gives the degenerate normal limit.

Facts & Assumptions

Given: AC, the centered strictly stationary real sequence $Y$ , finite second moment, and $\sum_{n \geq 1} ρ_{Y} (n) < \infty$ .

[A1]

AC is the inherited assumption of the local extension and martingale CLT suppliers. (The Axiom of Choice)

[F1]

The coefficient $ρ_{Y} (n)$ bounds absolute correlations between the finite past and separated future. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

The sequence has a two-sided stationary realization $X_{k} = X_{0} \circ T^{k}$ with the same nonnegative-index law and the same maximal-correlation upper bounds for the full past; if $ρ_{Y} (n) \to 0$ , its invertible shift is ergodic even for mod-null invariant events. (Two-sided stationary extension preserves maximal-correlation bounds)

[F3]

Conditional expectation is the $L^{2}$ orthogonal projection onto the conditioning subspace and preserves ordinary expectation. (Conditional expectation is the l2 orthogonal projection, Basic algebra and order properties of conditional expectation)

[F4]

For a centered $L^{2}$ process generated by an invertible probability-preserving ergodic transformation, summability of $∥ E [X_{j} ∣ F_{0}] ∥_{2}$ gives a normal CLT with variance $E [D_{1}^{2}] \geq 0$ , including zero; under absolute covariance summability this variance equals the covariance series. (Summable conditional projections give a stationary martingale CLT)

[F5]

Covariance and variance are centered product and square moments. (Moments, variance, and covariance on a probability space)

Proof

1.1A1F1F2given

Summability of the nonnegative numbers $ρ_{Y} (n)$ implies $ρ_{Y} (n) \to 0$ . By [F2], realize $Y$ as the nonnegative restriction in law of the two-sided stationary sequence $X_{k} = X_{0} \circ T^{k}$ with an invertible probability-preserving shift $T$ that is ergodic for mod-null invariant events. Its full past $F_{k} = σ (X_{i} : i \leq k)$ satisfies $∣ Cov (U, V) ∣ \leq ρ_{Y} (n) \sqrt{Var (U) Var (V)}$ for past $U$ and future $V$ separated by $n$ . All second moments of the coordinates agree with those of $Y$ .

2.1F3step 1.1givenalgebra

Put $W_{j} = E [X_{j} ∣ F_{0}]$ for $j \geq 1$ . By [F3], $W_{j}$ is centered and square-integrable and $E [W_{j} X_{j}] = E [W_{j}^{2}]$ . If $∥ W_{j} ∥_{2} = 0$ , the bound $∥ W_{j} ∥_{2} \leq ρ_{Y} (j) ∥ X_{0} ∥_{2}$ is immediate. Otherwise apply step 1.1's covariance inequality to $W_{j}$ and $X_{j}$ , divide by $∥ W_{j} ∥_{2}$ , and obtain that same bound. Consequently $\sum_{j \geq 1} ∥ E [X_{j} ∣ F_{0}] ∥_{2} \leq ∥ X_{0} ∥_{2} \sum_{j \geq 1} ρ_{Y} (j) < \infty$ .

3.1F4F5step 1.1step 2.1given∎

Put $γ_{j} = Cov (X_{0}, X_{j})$ . Step 1.1 applied to the two coordinates gives $∣ γ_{j} ∣ \leq ρ_{Y} (j) E [X_{0}^{2}]$ for $j \geq 1$ ; if $X_{0}$ is constant, both sides vanish. Thus $\sum_{j \geq 1} ∣ γ_{j} ∣ < \infty$ . All hypotheses of [F4] now hold by steps 1.1 and 2.1. It follows that $n^{- 1 / 2} \sum_{k = 1}^{n} X_{k} \Rightarrow N (0, τ^{2})$ and $τ^{2} = E [X_{0}^{2}] + 2 \sum_{j \geq 1} γ_{j} \geq 0$ . This includes $τ^{2} = 0$ and proves absolute covariance convergence. Since $X$ and $Y$ have the same nonnegative-index law, the series is exactly $σ_{Y}^{2}$ and the finite sums have the original laws. Therefore the asserted nondegenerate and degenerate limits both hold for $Y$ .

Source notes

The proof uses the stronger summable individual-projection criterion, proved locally by an exact martingale-coboundary series. No rho-mixing CLT is imported as a proof premise.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Strong Brownian approximation implies overlapping-batch consistency

Statement

Assume AC. Let $Y_{1}, Y_{2}, \dots$ be real random variables and $S_{k} = \sum_{i = 1}^{k} Y_{i}$ , with $S_{0} = 0$ . Suppose on a probability-space extension with the original process as marginal there are a standard Brownian motion $B$ , constants $σ \geq 0$ and $0 < α < 1 / 2$ , and finite random constants $K, N$ such that almost surely $∣ S_{k} - σ B (k) ∣ \leq K k^{α} \log k (k \geq N) .$ Let $(b_{n})$ be deterministic positive integers with $b_{n} < n$ eventually and $\frac{b_{n}}{n} \to 0, \sum_{n = 2}^{\infty} (b_{n} / n)^{c} < \infty for some real c \geq 1,$ $\frac{b_{n} \log n}{n} \to 0, \frac{n^{2 α} (\log n)^{2}}{b_{n}} \to 0.$ For the exact estimator of Overlapping batch means for MCMC standard errors, applied to $Y_{1}, \dots, Y_{n}$ , ${\hat{σ}}_{O B M, n}^{2} = \frac{n b_{n}}{(n - b_{n}) (n - b_{n} + 1)} \sum_{j = 0}^{n - b_{n}} {(\frac{S_{j + b_{n}} - S_{j}}{b_{n}} - \frac{S_{n}}{n})}^{2} ⟶ σ^{2} almost surely .$ No stationarity, independence of $B$ from $(Y_{i})$ , or monotonicity of $b_{n}$ or $n / b_{n}$ is required. The conclusion includes $σ = 0$ .

Facts & Assumptions

Given: AC, the original process and its Brownian approximation on the stated extension, the constants $σ, α$ , and the deterministic batch sequence with all displayed numerical hypotheses.

[A1]

AC is assumed as in The Axiom of Choice and the Brownian and Gaussian interfaces.

[F1]

Brownian motion starts at zero and has independent normal increments with variance equal to interval length. Brownian motion

[F2]

A centered Gaussian quadratic form satisfies the dimension-free even-moment bound for every positive integer $r$ . Centered Gaussian quadratic forms have dimension-free even-moment bounds

[F3]

For a nonnegative random variable $X$ and $a > 0$ , $\Pr (X \geq a) \leq E X / a$ . Markov's inequality for random variables

[F4]

Events with summable probabilities occur only finitely often almost surely. The first Borel-Cantelli lemma for measures

[F5]

The standard-normal Mills upper bound $\Pr (Z > x) \leq e^{- x^{2} / 2} / (x \sqrt{2 π})$ , together with symmetry and the normal density, gives $\Pr (∣ Z ∣ \geq x) \leq 2 e^{- x^{2} / 2}$ for $x \geq 1$ . Two-sided Mills bounds for the standard normal tail

[F6]

The Euclidean norm is a norm and satisfies $∣ ∥ x ∥_{2} - ∥ y ∥_{2} ∣ \leq ∥ x - y ∥_{2}$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, The reverse triangle inequality in a normed space

[F7]

The finite OBM algorithm uses the displayed denominator and overlapping batch means. Overlapping batch means for MCMC standard errors

Proof

1.1A1F1givenconstructalgebra

Work first on the extension. For each sufficiently large $n$ write $b = b_{n}$ , $m = n - b + 1$ , $U_{i} = B (i) - B (i - 1)$ for $1 \leq i \leq n$ , and $D_{j} = B (j + b) - B (j)$ for $0 \leq j \leq n - b$ . By [F1], the $U_{i}$ are independent standard normals. If $h_{j}$ indicates coordinates $j + 1, \dots, j + b$ , then $Q_{n} = (n b)^{- 1} \sum_{j} D_{j}^{2} = U^{T} A_{n} U$ where $A_{n} = (n b)^{- 1} \sum_{j} h_{j} h_{j}^{T}$ . At most $b$ windows contain a fixed pair of coordinates, and none does if their separation is at least $b$ . Hence $0 \leq (A_{n})_{i k} \leq 1 / n$ , $(A_{n})_{i k} = 0$ when $∣ i - k ∣ \geq b$ , $tr A_{n} = m / n$ , and $\sum_{i, k} (A_{n})_{i k}^{2} \leq (2 b - 1) / n \leq 2 b / n$ .

2.1F2F3F4step 1.1givenchoosealgebra

Choose an integer $r \geq c$ . By [F2] and [F3], for every $η > 0$ , $\Pr (∣ Q_{n} - m / n ∣ > η) \leq C_{r} η^{- 2 r} (2 b_{n} / n)^{r}$ . These probabilities are summable since $b_{n} / n < 1$ eventually, $r \geq c$ and $\sum_{n} (b_{n} / n)^{c} < \infty$ . Applying [F4] for each $η = 1, 1 / 2, 1 / 3, \dots$ and intersecting the resulting countable family of full-probability events gives $Q_{n} - m / n \to 0$ almost surely. Since $m / n \to 1$ , $Q_{n} \to 1$ almost surely. No independence between different $n$ is used.

3.1F1F4F5F6step 1.1step 2.1givenalgebra

By [F1] and [F5], $\Pr (∣ B (n) ∣ > \sqrt{6 n \log n}) \leq 2 n^{- 3}$ for $n \geq 2$ . Thus [F4] gives $B (n)^{2} / n \leq 6 \log n$ eventually almost surely. Put $V_{j} = D_{j} - (b / n) B (n)$ and $W_{n} = (n b)^{- 1} \sum_{j} V_{j}^{2}$ . Applying [F6] to the vectors $(V_{j}) / \sqrt{n b}$ and $(D_{j}) / \sqrt{n b}$ gives $∣ \sqrt{W_{n}} - \sqrt{Q_{n}} ∣ \leq [(m / n) (b / n) (B (n)^{2} / n)]^{1 / 2} \to 0$ , using $(b_{n} / n) \log n \to 0$ . Consequently $W_{n} \to 1$ almost surely.

4.1step 3.1givenconstructalgebra

Set $R_{k} = S_{k} - σ B (k)$ and $R_{0} = 0$ . On the full event of the given approximation, enlarge its finite random constant to absorb all finitely many early errors. Since $k^{α} \log k$ is increasing for integers $k \geq 2$ , this gives a finite random $K^{'}$ with $\max_{0 \leq k \leq n} ∣ R_{k} ∣ \leq K^{'} n^{α} \log n$ for every sufficiently large $n$ . For every window, $S_{j + b} - S_{j} - (b / n) S_{n} = σ V_{j} + E_{n, j}$ where $E_{n, j} = R_{j + b} - R_{j} - (b / n) R_{n}$ . Therefore $∣ E_{n, j} ∣ \leq 3 K^{'} n^{α} \log n$ simultaneously for all windows, and $H_{n} = (n b)^{- 1} \sum_{j} E_{n, j}^{2} \leq 9 (K^{'})^{2} (m / n) n^{2 α} (\log n)^{2} / b_{n} \to 0$ .

5.1F6step 3.1step 4.1givenalgebra

Put $T_{n} = (n b)^{- 1} \sum_{j} (S_{j + b} - S_{j} - (b / n) S_{n})^{2}$ . By [F6] and $σ \geq 0$ , $∣ \sqrt{T_{n}} - σ \sqrt{W_{n}} ∣ \leq \sqrt{H_{n}} \to 0$ . Hence $T_{n} \to σ^{2}$ almost surely. This argument divides neither by $σ$ nor by $T_{n}$ , so it also proves the zero-variance case.

6.1F7step 5.1givenalgebra∎

By [F7], ${\hat{σ}}_{O B M, n}^{2} = n^{2} T_{n} / [(n - b_{n}) (n - b_{n} + 1)]$ . This deterministic multiplier tends to one because $b_{n} / n \to 0$ , and step 5.1 proves the desired limit on the extension. The event that this sequence of finite measurable functions of $(Y_{i})$ converges to $σ^{2}$ depends only on the original process. Its probability under the extension equals its probability under the original marginal, so that probability is one for the original process as well. Arbitrarily defining the estimator at any finitely many indices with inadmissible batches does not affect its limit.

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Bounded stationary CLT under summable strong mixing

Statement

Assume AC. Let $(Z_{k})_{k \geq 0}$ be a centered strictly stationary real sequence, with $∣ Z_{0} ∣ \leq B$ almost surely for a finite $B$ , and suppose $\sum_{r = 1}^{\infty} α_{Z} (r) < \infty .$ Put $T_{n} = \sum_{k = 1}^{n} Z_{k}$ . Then its covariance series converges absolutely and $σ_{Z}^{2} = E Z_{0}^{2} + 2 \sum_{r = 1}^{\infty} Cov (Z_{0}, Z_{r}) = \lim_{n \to \infty} \frac{Var (T_{n})}{n} \geq 0.$ Moreover $T_{n} / \sqrt{n} \Rightarrow N (0, σ_{Z}^{2}),$ including $σ_{Z}^{2} = 0$ . Writing $D_{Z} = E Z_{0}^{2} + 2 \sum_{r \geq 1} ∣ Cov (Z_{0}, Z_{r}) ∣$ , every finite subset $I \subseteq N_{0}$ satisfies $E {(\sum_{i \in I} Z_{i})}^{2} \leq D_{Z} ∣ I ∣ .$ The strong-mixing coefficient is that of Strong, uniform, and maximal-correlation mixing coefficients.

Facts & Assumptions

Given: AC, the centered strictly stationary bounded sequence, and summability of its strong-mixing coefficients.

[A1]

AC is assumed for the characteristic-function convergence criterion and the normal-law supplier. (The Axiom of Choice)

[F1]

Strict stationarity preserves finite-dimensional laws under nonnegative shifts, and the mixing coefficient is the supremum over separated past and future events. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

Bounded real variables from fields with event coefficient $a$ have covariance at most $4 A B a$ in absolute value; complex variables obey the analogous bound $16 A B a$ . (Strong-mixing covariance bound in terms of tail quantiles)

[F3]

Variance and covariance are centered product expectations. (Moments, variance, and covariance on a probability space)

[F4]

Dominated convergence applies on an arbitrary measure space, including counting measure with a summable majorant. (Dominated convergence)

[F5]

Cauchy–Schwarz bounds absolute product expectations of square-integrable real variables. (Cauchy-Schwarz for random variables)

[F6]

For $r (u) = e^{i u} - 1 - i u + u^{2} / 2$ , one has $∣ r (u) ∣ \leq ∣ u ∣^{3} / 3$ for all real $u$ . (Second-order characteristic-function expansion)

[F7]

Under AC the characteristic function of $N (0, s^{2})$ is $e^{- s^{2} t^{2} / 2}$ , including $s = 0$ . (Characteristic function of a normal law)

[F8]

Under AC, convergence of characteristic functions to that of a specified probability law is equivalent to weak convergence to that law. (Characteristic function criterion for weak convergence)

Proof

1.1F1F2F3F4given

Put $γ_{r} = Cov (Z_{0}, Z_{r})$ . By [F1]–[F2], $∣ γ_{r} ∣ \leq 4 B^{2} α_{Z} (r)$ for $r \geq 1$ , so $D_{Z} < \infty$ . Expanding the square of a sum indexed by a finite subset $I$ gives its diagonal terms $∣ I ∣ γ_{0}$ and off-diagonal terms $2 γ_{j - i}$ for pairs $i < j$ in $I$ . For each fixed positive lag there are at most $∣ I ∣$ such pairs. Thus the asserted bound $E (\sum_{i \in I} Z_{i})^{2} \leq D_{Z} ∣ I ∣$ holds, also for $I = \emptyset$ . For a consecutive block, the same expansion gives $Var (T_{n}) / n = γ_{0} + 2 \sum_{r = 1}^{n - 1} (1 - r / n) γ_{r}$ . The summable majorant $2 ∣ γ_{r} ∣$ and [F4] imply convergence to the stated $σ_{Z}^{2}$ , which is nonnegative as a limit of variances.

1.2F1F2givenalgebra

Extend the coefficient convention by $α_{Z} (0) = 1$ . It is nonincreasing: increasing the separation restricts the future sigma-algebra in [F1]. For ordered indices $i \leq j \leq k \leq l$ , write $a = j - i$ and $c = l - k$ . If $a > 0$ , centering of $Z_{i}$ and [F2] applied to $Z_{i}$ and $Z_{j} Z_{k} Z_{l}$ give $∣ E Z_{i} Z_{j} Z_{k} Z_{l} ∣ \leq 4 B^{4} α_{Z} (a)$ ; if $a = 0$ , the trivial bound $B^{4}$ suffices. Splitting off the centered last factor gives the same bound with $c$ . Therefore $∣ E Z_{i} Z_{j} Z_{k} Z_{l} ∣ \leq 4 B^{4} α_{Z} (\max (a, c))$ . There are at most $24$ orderings, at most $p$ choices each for the first index and the middle gap, and $2 r + 1$ pairs $(a, c)$ with maximum $r$ . Hence $E T_{p}^{4} \leq 96 B^{4} p^{2} \sum_{r = 0}^{p} (2 r + 1) α_{Z} (r) = o (p^{3}) .$ To justify the last limit, for each fixed $R$ , $p^{- 1} \sum_{r \leq R} r α_{Z} (r) \to 0$ , whereas $p^{- 1} \sum_{R < r \leq p} r α_{Z} (r) \leq \sum_{r > R} α_{Z} (r) \to 0$ as $R \to \infty$ ; the unweighted part divided by $p$ also tends to zero. Repeated indices are included by the convention at zero.

2.1step 1.1step 1.2algebra

Set $ε_{p} = E T_{p}^{4} / p^{3}$ and $η_{n} = \sup_{p \geq ⌊ \sqrt{n} ⌋} ε_{p}$ . By step 1.2, $η_{n} \to 0$ . For sufficiently large $n$ , define $h_{n} = \min (n^{1 / 8}, η_{n}^{- 1 / 4})$ , interpreting the second entry as $+ \infty$ if $η_{n} = 0$ , and ignore the finitely many earlier $n$ . Then $h_{n} \to \infty$ , $h_{n} = o (n^{1 / 4})$ , and $h_{n}^{2} η_{n} \to 0$ . Put $p_{n} = ⌈ \sqrt{n} h_{n} ⌉$ , $q_{n} = ⌊ \sqrt{n} / h_{n} ⌋$ , and $k_{n} = ⌊ n / (p_{n} + q_{n}) ⌋$ . Thus $p_{n}, q_{n} \to \infty$ , $p_{n} / n \to 0$ , $q_{n} / p_{n} \to 0$ , and $k_{n} p_{n} / n \to 1$ . Retain $k_{n}$ consecutive blocks of length $p_{n}$ , separated by discarded blocks of length $q_{n}$ , and discard the final remainder. Their discarded index set $I_{n}$ has cardinality at most $k_{n} q_{n} + p_{n} + q_{n} = o (n)$ . Step 1.1 gives $∥ n^{- 1 / 2} \sum_{i \in I_{n}} Z_{i} ∥_{2}^{2} \leq D_{Z} ∣ I_{n} ∣ / n \to 0$ , controlling all gaps together without asserting their independence.

3.1F1F2step 2.1algebra

Summability and monotonicity imply $r α_{Z} (r) \to 0$ , since $⌊ r / 2 ⌋ α_{Z} (r) \leq \sum_{s = ⌊ r / 2 ⌋}^{r} α_{Z} (s)$ . For large $n$ , $n / p_{n} \leq \sqrt{n} / h_{n} \leq 2 q_{n}$ , so $k_{n} α_{Z} (q_{n}) \leq 2 q_{n} α_{Z} (q_{n}) \to 0$ . Let $B_{n, j}$ denote the sum in retained block $j$ . At a fixed real frequency $t$ , put $U_{j} = e^{i t B_{n, j} / \sqrt{n}}$ . Both each factor and every partial product have modulus at most one. The partial product through block $j$ is measurable before the gap preceding block $j + 1$ , so [F2] bounds the discrepancy in factoring its expectation from $U_{j + 1}$ by $16 α_{Z} (q_{n})$ . Telescoping these factorizations gives $∣ E e^{i t \sum_{j = 1}^{k_{n}} B_{n, j} / \sqrt{n}} - \prod_{j = 1}^{k_{n}} E e^{i t B_{n, j} / \sqrt{n}} ∣ \leq 16 (k_{n} - 1) α_{Z} (q_{n}) ⟶ 0.$ This compares characteristic functions only and requires no independent copies or coupling construction.

3.2F1F5step 1.1step 2.1algebra

Write $W_{n, j} = B_{n, j} / \sqrt{n}$ . These variables are centered and each has the law of $T_{p_{n}} / \sqrt{n}$ by stationarity. Steps 1.1 and 2.1 imply $\sum_{j = 1}^{k_{n}} E W_{n, j}^{2} = (k_{n} p_{n} / n) Var (T_{p_{n}}) / p_{n} \to σ_{Z}^{2}$ . Also $\sum_{j = 1}^{k_{n}} E W_{n, j}^{4} = \frac{k_{n} E T_{p_{n}}^{4}}{n^{2}} \leq \frac{p_{n}^{2}}{n} ε_{p_{n}} ⟶ 0,$ because $p_{n}^{2} / n = O (h_{n}^{2})$ and $ε_{p_{n}} \leq η_{n}$ . For every $δ > 0$ , the Lindeberg sum of these block marginals is at most $δ^{- 2} \sum_{j} E W_{n, j}^{4} \to 0$ . Applying [F5] first to $∣ W ∣, W^{2}$ and then the finite-sum Cauchy–Schwarz inequality yields $\sum_{j} E ∣ W_{n, j} ∣^{3} \leq (\sum_{j} E W_{n, j}^{2})^{1 / 2} (\sum_{j} E W_{n, j}^{4})^{1 / 2} \to 0$ . Finally $\max_{j} (E W_{n, j}^{2})^{2} \leq \sum_{j} E W_{n, j}^{4} \to 0$ . No uniform-integrability hypothesis or deterministic block bound tending to zero has been assumed.

4.1F6step 3.2algebra

For fixed $t$ , [F6] gives $E e^{i t W_{n, j}} = 1 - a_{n, j} + r_{n, j}$ , where $a_{n, j} = t^{2} E W_{n, j}^{2} / 2 \geq 0$ and $\sum_{j} ∣ r_{n, j} ∣ \leq ∣ t ∣^{3} \sum_{j} E ∣ W_{n, j} ∣^{3} / 3 \to 0$ . Step 3.2 gives $\sum_{j} a_{n, j} \to t^{2} σ_{Z}^{2} / 2$ and $\sum_{j} a_{n, j}^{2} \leq (\max_{j} a_{n, j}) \sum_{j} a_{n, j} \to 0$ . The elementary real Taylor bound $∣ 1 - a - e^{- a} ∣ \leq a^{2} / 2$ for $a \geq 0$ follows by integrating the second derivative $e^{- a} \leq 1$ twice. Since both $∣ E e^{i t W_{n, j}} ∣$ and $e^{- a_{n, j}}$ are at most one, the product telescoping inequality now gives $∣ \prod_{j} E e^{i t W_{n, j}} - e^{- \sum_{j} a_{n, j}} ∣ \leq \sum_{j} ∣ r_{n, j} ∣ + \frac{1}{2} \sum_{j} a_{n, j}^{2} ⟶ 0.$ Thus the product characteristic function tends to $e^{- t^{2} σ_{Z}^{2} / 2}$ .

5.1A1F5F7F8step 1.1step 2.1step 3.1step 4.1∎

The discarded normalized sum in step 2.1 tends to zero in $L^{2}$ and hence in $L^{1}$ by [F5]. The elementary bound $∣ e^{i x} - e^{i y} ∣ \leq ∣ x - y ∣$ , obtained by integrating the derivative of $e^{i u}$ along the real interval, makes its effect on each characteristic function tend to zero. Combine steps 3.1 and 4.1: $E e^{i t T_{n} / \sqrt{n}} \to e^{- t^{2} σ_{Z}^{2} / 2}$ for every real $t$ . Under [A1], [F7] identifies this with the characteristic function of $N (0, σ_{Z}^{2})$ , and [F8] gives the asserted weak convergence. This remains valid for $σ_{Z}^{2} = 0$ ; all finite-moment and discarded-set assertions were already proved in step 1.1.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Geometric strong mixing gives an Lp Poisson solution

Statement

Assume AC. Let $P$ be a probability kernel on an arbitrary measurable space $(X, X)$ with invariant probability $π$ . Let $(X_{k})_{k \geq 0}$ be its stationary Markov chain, and suppose its strong-mixing coefficients satisfy $α_{X} (n) \leq C r^{n}$ for $n \geq 1$ , with $0 \leq C < \infty$ and $0 < r < 1$ . Let $f : X \to R$ be measurable with $π f = 0$ and $π ∣ f ∣^{q} < \infty$ for some finite $q > 2$ .

For every $2 < p < q$ , put $θ = 1 / p - 1 / q$ . The kernel operators are understood on $π$ -almost-everywhere classes, with finite measurable versions obtained by setting divergent integrals to zero on their $π$ -null exceptional sets. Then $∥ P^{n} f ∥_{L^{p} (π)} \leq 4 α_{X} (n)^{θ} ∥ f ∥_{L^{q} (π)} (n \geq 1),$ and the series $g = \sum_{n = 0}^{\infty} P^{n} f$ converges in $L^{p} (π)$ , with $f = g - P g π -almost everywhere .$ Moreover, writing $c_{n} = {Cov}_{π} (f (X_{0}), f (X_{n}))$ , one has $∣ c_{n} ∣ \leq 4 α_{X} (n)^{1 - 2 / q} ∥ f ∥_{L^{q} (π)}^{2} (n \geq 1), \sum_{n \geq 1} ∣ c_{n} ∣ < \infty .$ The covariance sum $v = π (f^{2}) + 2 \sum_{n \geq 1} c_{n}$ is finite and nonnegative, and $\frac{1}{n} E_{π} {(\sum_{k = 0}^{n - 1} f (X_{k}))}^{2} ⟶ v .$

Facts & Assumptions

Given: AC, the stationary kernel and invariant probability, the geometric strong-mixing bound, centered real $f \in L^{q} (π)$ , and $2 < p < q$ .

[A1]

AC is assumed. The Axiom of Choice

[F1]

For square-integrable variables in sigma-algebras with event mixing coefficient $a$ , $∣ Cov (U, V) ∣ \leq 4 \int_{0}^{a} Q_{U} Q_{V}$ , and $\int_{0}^{1} ψ (Q_{U} (u)) d u = E ψ (∣ U ∣)$ for nonnegative Borel $ψ$ . Separation by $n$ bounds $a$ by $α_{X} (n)$ . Strong-mixing covariance bound in terms of tail quantiles, Strong, uniform, and maximal-correlation mixing coefficients

[F2]

The bounded Markov identity is $E [h (X_{j + n}) ∣ F_{j}] = P^{n} h (X_{j})$ ; its square-integrable extension identifies the conditional mean with the kernel integral on a full invariant-measure set. Chapman-Kolmogorov equations, Kernel conditional expectations for square-integrable Markov observables

[F3]

Conditional expectation is an $L^{p}$ contraction, and on a probability space $L^{q} \subset L^{p} \subset L^{2}$ when $q > p > 2$ . Conditional lp contraction, Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$

[F4]

Hölder and its product form bound products in the corresponding $L^{s}$ spaces; Minkowski bounds the norm of a finite sum by the sum of norms. Holder's inequality for integrals, including the endpoint cases, Generalized Holder inequality puts products into $L^{r}$ , Minkowski's inequality for integrals, including $p = \infty$

[F5]

Under AC, $L^{p}$ is complete, consists of almost-everywhere classes of measurable representatives, and norm-convergent sequences have almost-everywhere convergent representative subsequences. Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ , The space $L^{p} (μ)$ as the quotient by null functions

[F6]

Increasing nonnegative measurable functions pass through integration by monotone convergence. Monotone convergence for the integral

[F7]

Invariance means $π P = π$ ; the identity extends from indicators to nonnegative measurable functions by increasing simple approximations. Invariant and stationary distribution for a Markov kernel

[F8]

Centered second moments are variances, covariance is the centered product expectation, and covariance is bilinear and symmetric. Moments, variance, and covariance on a probability space, Covariance is symmetric and bilinear in finite linear combinations

Proof

1.1A1F2F3F6F7given

Invariance [F7], extended with [F6], gives $π P ∣ h ∣ = π ∣ h ∣$ for integrable $h$ , so the absolute kernel integral is finite off a measurable $π$ -null set. A change of representative on a $π$ -null set $A$ does not change its image class, since $π P 1_{A} = π (A) = 0$ implies $P 1_{A} = 0$ almost everywhere. For $h \in L^{p}$ , [F3] gives $h \in L^{2}$ , and [F2] identifies $P h (X_{0})$ with $E [h (X_{1}) ∣ X_{0}]$ . Conditional contraction [F3] and stationarity give $∥ P h ∥_{p} \leq ∥ h ∥_{p}$ . The bounded iteration identity in [F2] extends to these $L^{p}$ classes by replacing $h$ with its bounded clipped truncations: their $L^{p}$ error tends to zero because the integrable tail $∣ h ∣^{p} 1_{{∣ h ∣ > K}}$ tends to zero, as follows from [F6] applied to its complementary truncations; conditional and kernel contractions pass this error to zero. Thus $P$ is a well-defined linear contraction on $L^{p}$ , its iterates agree with the kernel iterates, and $P^{n} h (X_{0}) = E [h (X_{n}) ∣ X_{0}]$ . This also proves $P^{n} f \in L^{p}$ before any covariance estimate is used.

2.1F1F3step 1.1givenconstructalgebra

Fix $n \geq 1$ , put $Z = P^{n} f$ , and let $p^{'} = p / (p - 1)$ and $W_{K} (x) = sgn (Z (x)) ∣ Z (x) ∣^{p - 1} 1_{{∣ Z (x) ∣ \leq K}}$ for integers $K \geq 1$ . This function is bounded, belongs to $L^{2}$ , and satisfies $∥ W_{K} ∥_{p^{'}} \leq ∥ Z ∥_{p}^{p - 1}$ . The centered $f (X_{n})$ is in $L^{2}$ by [F3]. Step 1.1 and bounded conditional multiplication give $E [W_{K} (X_{0}) f (X_{n})] = π (W_{K} P^{n} f) = π (∣ Z ∣^{p} 1_{{∣ Z ∣ \leq K}})$ ; since $π f = 0$ , this expectation is the covariance of $W_{K} (X_{0})$ and $f (X_{n})$ . Bounded conditional multiplication here follows directly from the defining conditional expectation event identity, first for simple $W_{K}$ and then by uniform bounded approximation. Thus [F1] is applied only to square-integrable variables, even though $p^{'} < 2$ .

3.1F1F4F6step 2.1algebra

Put $a = α_{X} (n)$ . The quantile identity in [F1] gives $∥ Q_{W_{K} (X_{0})} ∥_{p^{'}} = ∥ W_{K} ∥_{p^{'}}$ and $∥ Q_{f (X_{n})} ∥_{q} = ∥ f ∥_{q}$ . Since $1 / p^{'} + 1 / q = 1 - θ$ , apply the product inequality [F4] to these two quantile functions in $L^{1 / (1 - θ)} (0, 1)$ , and then Hölder with $1_{(0, a)}$ , whose $L^{1 / θ}$ norm is $a^{θ}$ . Therefore [F1] and step 2.1 give $π (∣ Z ∣^{p} 1_{{∣ Z ∣ \leq K}}) \leq 4 a^{θ} ∥ Z ∥_{p}^{p - 1} ∥ f ∥_{q}$ . If $a = 0$ the quantile integral and the right-hand side are zero. Let $K \to \infty$ using [F6]. If $∥ Z ∥_{p} = 0$ the required bound is immediate; otherwise divide by $∥ Z ∥_{p}^{p - 1}$ to obtain $∥ P^{n} f ∥_{p} \leq 4 α_{X} (n)^{θ} ∥ f ∥_{q}$ . The restriction $p < q$ ensures $θ > 0$ and makes all Hölder exponents just used valid.

4.1A1F4F5step 1.1step 3.1constructalgebra

The geometric bound makes $\sum_{n \geq 1} α_{X} (n)^{θ} \leq C^{θ} \sum_{n \geq 1} r^{n θ} < \infty$ , including $C = 0$ . By step 3.1, [F4] makes $g_{m} = \sum_{n = 0}^{m} P^{n} f$ a Cauchy sequence in $L^{p}$ . Completeness [F5] gives $g_{m} \to g \in L^{p}$ . The contraction in step 1.1 gives $P g_{m} \to P g$ , while $g_{m} - P g_{m} = f - P^{m + 1} f \to f$ in $L^{p}$ by step 3.1. Thus $g - P g = f$ as almost-everywhere classes. Choose the finite measurable representative supplied by [F5], setting it to zero on its measurable null exceptional set if necessary, and use the version convention from step 1.1 for $P g$ . These choices make the asserted almost-everywhere identity precise on the original sigma-algebra without completing the measure.

4.2F1F3F4step 3.1givenalgebra

Apply [F1] directly to $f (X_{0}), f (X_{n})$ , both square-integrable. Their quantiles have common $L^{q} (0, 1)$ norm $∥ f ∥_{q}$ . Two applications of [F4], with reciprocal exponents $1 / q, 1 / q, 1 - 2 / q$ , yield $∣ c_{n} ∣ \leq 4 α_{X} (n)^{1 - 2 / q} ∥ f ∥_{q}^{2}$ . The exponent is positive because $q > 2$ ; the geometric bound therefore gives $\sum_{n \geq 1} ∣ c_{n} ∣ < \infty$ . At a zero mixing coefficient the quantile integral is zero, so no division by that coefficient is needed.

5.1A1F5F8step 4.1step 4.2givenalgebra∎

Centering, stationarity and [F8] give $n^{- 1} E_{π} (\sum_{k = 0}^{n - 1} f (X_{k}))^{2} = π (f^{2}) + 2 \sum_{h = 1}^{n - 1} (1 - h / n) c_{h}$ . To pass to its limit, choose any finite $H$ ; the terms $h \leq H$ converge individually to $c_{h}$ , while the absolute value of either remaining weighted or limiting tail is at most $\sum_{h > H} ∣ c_{h} ∣$ , which tends to zero as $H \to \infty$ by step 4.2. The limit is the claimed finite covariance sum $v$ . Each left-hand side is nonnegative, hence $v \geq 0$ . When $f = 0$ almost everywhere all bounds and the choice $g = 0$ hold directly; no claim of uniqueness of $g$ is made. The strict exponent range and all zero-mixing cases were handled in steps 3.1, 4.1, 4.2. AC is used through the Markov/conditional and completeness suppliers, not as an imported covariance or Poisson theorem.

Source notes

The actual Doukhan–Massart–Rio full PDF was retrieved and its printed p. 66 inspected as a rendered page, including the finite-variance restriction of Lemma 1 and the covariance-sum conclusion of Proposition 1. The proof above consumes the fully proved local quantile lemma. Its test variable is bounded before that lemma is applied; the unbounded $L^{p^{'}}$ dual variable, whose exponent is less than two, is never inserted into a result requiring square integrability. The Poisson-series construction, norm convergence and nonnegative variance limit are proved here, not replaced by a source citation.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Conditional quantile couplings for Gaussian anchored sums

Statement

Assume AC and the hypotheses of Gaussian anchored bounded sums have uniform cumulant and Fourier bounds: $S = Z + \sum_{i = 1}^{m} X_{i} \in R^{d}$ , $1 \leq d \leq 4$ , the centered summands are independent, $∣ X_{i} ∣ \leq B$ , $Z$ is Gaussian with covariance $G \geq γ s I_{d}$ , and $cov (S) = s I_{d}$ , where $s, B > 0$ and $0 < γ \leq 1$ . Put $τ = C_{0} B$ , with $C_{0}$ from that supplier, and let $Φ_{s}$ be the CDF of $N (0, s)$ . There are constants $c_{γ} > 0$ and $C_{γ} < \infty$ , independent of the summand laws, $m, d, s, B$ , with the following properties.

For $1 \leq j \leq d$ , let $H_{j} (x^{'}, y)$ be the conditional CDF of $S_{j}$ given $S_{< j} = x^{'} \in R^{j - 1}$ obtained by dividing the joint prefix density by its marginal density. These densities are strictly positive, so this defines a probability kernel at every $x^{'}$ , including prefixes of probability zero. Its quantile transport $T_{j} (x^{'}, z) = H_{j} (x^{'}, \cdot)^{- 1} (Φ_{s} (z))$ is jointly Borel measurable and takes a fresh $N (0, s)$ variable to that conditional law. If $s \geq C_{γ} τ^{2}$ , then, with $y = T_{j} (x^{'}, z)$ , $∣ y - z ∣ \leq C_{γ} τ (1 + \frac{∣ x^{'} ∣^{2} + y^{2}}{s}) when ∣ x^{'} ∣ + ∣ y ∣ \leq c_{γ} s / τ,$ and also $∣ T_{j} (x^{'}, z) - z ∣ \leq C_{γ} τ (1 + \frac{∣ x^{'} ∣^{2} + z^{2}}{s}) when ∣ x^{'} ∣ + ∣ z ∣ \leq c_{γ} s / τ .$ For $j = 1$ , the prefix is empty and its norm is zero. The resulting successive coordinate transport $R$ sends $N_{d} (0, s I_{d})$ to the law of $S$ and satisfies $∣ R (g) - g ∣ \leq C_{γ} τ (1 + \frac{∣ R (g) ∣^{2}}{s}) when ∣ R (g) ∣ \leq c_{γ} s / τ .$

If $d = 2 a$ with $a \leq 2$ , write $S = (P, D)$ with $P, D \in R^{a}$ . The last $a$ successive conditional transports give a jointly measurable $T_{p}$ taking a fresh $E \sim N_{a} (0, s I_{a})$ to the conditional law of $D$ given $P = p$ , and $∣ T_{p} (e) - e ∣ \leq C_{γ} τ (1 + \frac{∣ p ∣^{2} + ∣ T_{p} (e) ∣^{2}}{s})$ when $s \geq C_{γ} τ^{2}$ and $∣ p ∣, ∣ T_{p} (e) ∣ \leq c_{γ} s / τ$ . The Gaussian reference bridge $E$ is independent of its Gaussian parent.

There is also the following scalar unequal-variance form. Let $L, R$ be independent centered scalar sums of independent variables bounded by $B$ and independent Gaussian terms, with variances $V_{L}, V_{R} \in [s / 2, s]$ , and with Gaussian component variances at least $γ V_{L}, γ V_{R}$ , respectively. Put $W = V_{L} + V_{R}, w = V_{L} / W, v = V_{L} V_{R} / W, P = L + R, D = L - w P .$ Then $W \in [s, 2 s]$ , $w \in [1 / 3, 2 / 3]$ , $v \in [s / 4, s / 2]$ . There is a jointly measurable transport taking a fresh $N (0, v)$ variable to $D$ conditional on $P = p$ with $∣ T_{p} (e) - e ∣ \leq C_{γ} τ (1 + \frac{p^{2} + T_{p} (e)^{2}}{s})$ when $s \geq C_{γ} τ^{2}$ and $∣ p ∣, ∣ T_{p} (e) ∣ \leq c_{γ} s / τ$ . The corresponding root transport from $N (0, W)$ to $P$ obeys the analogous bound with $∣ P ∣^{2} / s$ . The reconstructed children are $L = w P + D$ and $R = (1 - w) P - D$ .

All these maps are jointly measurable in an additional measurable parameter whenever, on each fixed finite row length, the bounded input laws are probability kernels, their Gaussian covariance and $s$ are measurable functions, and the displayed hypotheses hold parameterwise. This includes countable arrays with different row lengths. No claim here asserts the conditional estimates for raw bounded laws without a positive Gaussian component.

Facts & Assumptions

Given: The centered independent Gaussian anchored sum and its parameters as in the Statement; vector norms are Euclidean, and all constants below depend only on $γ$ .

[A1]

AC is assumed and inherited by the Gaussian suppliers. (The Axiom of Choice)

[F1]

The sum has an analytic cumulant with $∣ K_{u v v} (z) ∣ \leq τ s ∣ u ∣ ∣ v ∣^{2}$ , and every real tilt retains an independent Gaussian component with covariance at least $γ s I$ . Each coordinate prefix has the same properties. (Gaussian anchored bounded sums have uniform cumulant and Fourier bounds)

[F2]

Every tilted density is positive and continuously differentiable, has supremum at most $(2 π γ s)^{- d / 2}$ , and on the small tilt and moderate deviation region has its Gaussian density times $\exp (O_{γ} ((τ / \sqrt{s}) (1 + r^{3})))$ . Tilted covariance lies between $7 s I / 8$ and $9 s I / 8$ . The assertions hold in each prefix's coordinates. (Moderate deviation density bounds for Gaussian anchored sums)

[F3]

Positive definite Gaussian laws have their positive quadratic-form densities, and the standard normal CDF is continuous, strictly increasing, with limits zero and one. (A positive-definite multivariate normal law has its Gaussian density, The standard normal CDF is continuous and strictly increasing)

[F4]

Nonnegative product integration and dominated convergence apply to the integrals below. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Dominated convergence)

[F5]

Integrating a nonnegative product-measurable function against a finite or uniformly sigma-finite kernel gives a measurable function, and finite products of probability kernels are probability kernels. (Measurability of integration against a kernel, Finite products of probability kernels are probability kernels)

[F6]

Integration of derivatives along real line segments gives the corresponding norm and Lipschitz bounds; Euclidean Cauchy sequences converge. (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ )

[F7]

The rationals are countable and dense in the reals. ( $Q$ is countably infinite, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable)

Proof

1.1A1F1F2F6givenconstruct

Work with a fixed prefix of length $j \leq 4$ and write its dimension as $d$ during the proof. Its law satisfies [F1]–[F2]. Set $σ = \sqrt{s}$ , $ϵ = τ / σ$ , and normalize coordinates by $σ$ . Its cumulant $f$ has mean zero, covariance $I$ , and third derivatives bounded by $ϵ ∣ u ∣ ∣ v ∣^{2}$ . Polarization and [F6], integrating along a real segment, give $∥ f^{''} (a) - I ∥ \leq ϵ ∣ a ∣$ and $∣ f^{'} (a) - a ∣ \leq ϵ ∣ a ∣^{2} / 2$ . Fix a prefix $x^{'}$ , put $b = x^{'} / σ$ , $δ = ∣ b ∣$ , and assume $ϵ δ$ sufficiently small. The map $a^{'} \mapsto b - (\nabla^{'} f (a^{'}, 0) - a^{'})$ sends $∣ a^{'} ∣ \leq 2 δ$ into itself and has Lipschitz constant at most $2 ϵ δ < 1 / 2$ . Its iterates from zero have geometrically decreasing consecutive distances, hence converge by [F6] to a solution of $\nabla^{'} f (a^{'}, 0) = b$ with $∣ a^{'} ∣ \leq 2 δ$ . For the empty prefix use $a^{'} = 0$ . Put $a = (a^{'}, 0)$ and let $y_{0} = σ \partial_{d} f (a)$ be the physical last mean under this prefix tilt. The last mixed covariance row at zero vanishes; integrating its Hessian deviation yields $∣ y_{0} ∣ \leq C τ δ^{2}$ . The physical tilt parameter is $h = a / σ$ , so $τ ∣ h ∣ \leq 2 ϵ δ$ can be kept below $1 / 8$ .

2.1F2F3F4step 1.1algebra

Let $p, p^{'}$ be the joint and prefix densities. The Gaussian convolution representations in [F2] and [F4] give $\int_{R} p (x^{'}, y) d y = p^{'} (x^{'})$ at every $x^{'}$ : integrate the Gaussian joint density over its last coordinate and then integrate the bounded-summand law. Both densities are everywhere positive. Thus $p (x^{'}, y) / p^{'} (x^{'})$ defines a probability density for each $x^{'}$ and, by integration, a version of the conditional kernel. Denote tilted densities by $\overset{ˉ}{p}, {\overset{ˉ}{p}}^{'}$ . Since $h_{d} = 0$ , their exponential factors cancel exactly, and the density $v$ of $S_{d} - y_{0}$ conditional on $S^{'} = x^{'}$ is $v (u) = p (x^{'}, u + y_{0}) / p^{'} (x^{'}) = \overset{ˉ}{p} (x^{'}, u + y_{0}) / {\overset{ˉ}{p}}^{'} (x^{'})$ . The centered tilted covariance $C$ satisfies $∥ C / s - I ∥ \leq C ϵ δ$ by step 1.1. Apply [F2] to its full density at $(0, u)$ and prefix density at zero. The ratio of the matching Gaussian densities is a centered normal density with variance $\det C / \det C^{'} = 1 / (C^{- 1})_{d d} = s (1 + O (ϵ δ))$ ; the determinant identity follows by eliminating the off-diagonal blocks in $C$ . Consequently $∣ \log \frac{v (u)}{ϕ_{s} (u)} ∣ \leq C ϵ (1 + δ (1 + \frac{u^{2}}{s}) + \frac{∣ u ∣^{3}}{s^{3 / 2}})$ for $∣ u ∣ \leq c s / τ$ . Here $ϕ_{s}$ is the density of $N (0, s)$ , and the small constants also ensure all [F2] moderate bounds for the full centered tilt.

3.1F1F2F3F6step 1.1step 2.1algebra

In standardized units set $g (t) = σ v (σ t)$ and let $ϕ$ be the standard normal density. Step 2.1 implies, on $∣ t ∣ \leq c / ϵ$ , $∣ \log (g (t) / ϕ (t)) ∣ \leq C ϵ (1 + δ + δ t^{2} + ∣ t ∣^{3})$ and hence $c_{1} e^{- C_{1} t^{2}} \leq g (t) \leq C_{1} e^{- c_{1} t^{2}}$ after reducing $ϵ$ and $ϵ δ$ . There is also a global bound $g (t) \leq C \exp [- c \min (t^{2}, ∣ t ∣ / ϵ)]$ . Indeed the standardized prefix-tilted law has prefix density at its mean bounded below by a positive constant by [F2]. Tilt its centered full law further by $q e_{d}$ , where $∣ q ∣ \leq c / ϵ$ . Its full density is bounded above by a constant by the global part of [F2], while integration of its bounded Hessian gives a centered log mgf at most $C q^{2}$ . The density tilt identity therefore gives $g (t) \leq C e^{C q^{2} - q t}$ at every real $t$ . Choosing $q = sign (t) \min (∣ t ∣ / (4 C), c / (2 ϵ))$ proves the bound. The combined real tilt remains in a fixed smaller analytic ball, where the Hessian bound from step 1.1 applies; the global density bound in [F2] itself permits every real tilt.

4.1F3step 3.1algebrachoose

We next convert the density estimate into a CDF displacement without a coupling theorem. Write $R (t) = ϵ (1 + δ + δ t + t^{2})$ for $t \geq 0$ . Choose a fixed large $T \geq 10$ and then a sufficiently large fixed $K$ . Put $r (t) = K R (t)$ and $b = c_{0} / ϵ$ , with fixed $c_{0}$ sufficiently small. Reduce the allowed bounds on $ϵ$ and $ϵ δ$ so that for $T / 2 \leq t \leq 2 b$ one has $0 \leq r (t) \leq t / 8$ and $0 \leq r^{'} (t) \leq 1 / 2$ ; this is possible because $R^{'} (t) \leq 2 R (t) / t$ for $t \geq 1$ and $r (t) / t \leq C K ϵ (1 + δ + t)$ . Also choose $3 b$ within the moderate interval of step 3.1. For $t \geq T$ , the maps $f_{\pm} (t) = t \pm r (t)$ are increasing and their arguments are comparable with $t$ . Since $(\log ϕ)^{'} (t) = - t$ , step 3.1 bounds the logarithmic density error at either perturbed argument by $C t R (t)$ , whereas the Gaussian log-density changes by at least a fixed positive multiple of $t r (t)$ in the favorable direction. The log Jacobians are at most $r^{'} (t)$ and at least $- 2 r^{'} (t)$ , respectively. Since $r^{'} (t) \leq 2 r (t) / t$ , first taking $T$ large and then $K$ large gives $g (f_{+} (t)) f_{+}^{'} (t) \leq ϕ (t)$ and $g (f_{-} (t)) f_{-}^{'} (t) \geq ϕ (t)$ . These comparisons follow by adding the three logarithmic contributions; the Jacobian contribution divided by $t r (t)$ is at most $4 / t^{2}$ , and the density-error contribution is at most $C / K$ .

5.1F3step 3.1step 4.1algebrachoose

The inverse equations and monotonicity of $r$ imply $f_{+}^{- 1} (z) \geq z - r (z)$ and $f_{-}^{- 1} (z) \leq z + r (2 z)$ in the range below. Substituting $u = f_{+} (t)$ and $u = f_{-} (t)$ in the inequalities of step 4.1 gives, for $2 T \leq z \leq b$ , $\int_{z}^{b} g (u) d u \leq \int_{z - r (z)}^{\infty} ϕ (u) d u, \int_{z}^{b} g (u) d u \geq \int_{z + r (2 z)}^{b} ϕ (u) d u .$ Only a smaller range $z \leq c_{1} / ϵ$ , with $c_{1} ≪ c_{0}$ , will be retained, so the lower integral has its endpoints in the stated order. The global tail estimate in step 3.1 and the Gaussian tail give $\int_{b}^{\infty} g + \int_{b}^{\infty} ϕ \leq C e^{- c b^{2}}$ : split the first integral at a constant multiple of $1 / ϵ$ and integrate the Gaussian and exponential parts. Enlarge the shifts in the displayed inequalities to $3 r (2 z)$ . Each extra Gaussian interval has mass at least $c R (z) e^{- C z^{2}} \geq c ϵ e^{- C z^{2}}$ . Choose $c_{1}$ so small that $C c_{1}^{2} < c c_{0}^{2} / 2$ . For sufficiently small $ϵ$ , uniformly for $2 T \leq z \leq c_{1} / ϵ$ , that interval mass exceeds the discarded tail, since $ϵ^{- 1} \exp (- (c c_{0}^{2} - C c_{1}^{2}) / ϵ^{2}) \to 0$ . Thus the full tails have the corresponding inequalities with shift $C R (z)$ ; here $R (2 z) \leq 4 R (z)$ . Reflecting $g, ϕ$ proves the negative-tail version as well.

6.1F3step 3.1step 5.1algebrachoose

At $z = 2 T$ step 5.1 bounds the tail-probability difference by $C R (2 T)$ . For $∣ z ∣ \leq 2 T$ , step 3.1 bounds the density-ratio error by $C a$ , where $a = ϵ (1 + δ)$ ; integrating over this fixed interval gives a CDF difference at most $C a$ . The Gaussian density is bounded below by a fixed positive constant on $[- 2 T - 1, 2 T + 1]$ , so a shift of length $C a$ in either direction dominates that difference, after reducing the allowed bound on $a$ to keep the shift within this compact interval. Since $R (∣ z ∣) \geq a$ , this proves together with step 5.1, for the standardized conditional CDF $F_{g}$ , $Φ (z - C R (∣ z ∣)) \leq F_{g} (z) \leq Φ (z + C R (∣ z ∣)) (∣ z ∣ \leq c / ϵ) .$ All constant choices are finite and made in the indicated order; reducing the common cutoff enforces every smallness requirement. Returning to physical units, the CDF $F_{v}$ of the centered conditional coordinate satisfies $Φ_{s} (u - ρ (u)) \leq F_{v} (u) \leq Φ_{s} (u + ρ (u))$ , where $ρ (u) = C τ (1 + δ (1 + ∣ u ∣ / σ) + u^{2} / s)$ .

7.1F3step 1.1step 2.1step 6.1algebra

Every conditional density is positive and continuous by step 2.1, so its CDF is continuous, strictly increasing and has limits zero and one. Its inverse on $(0, 1)$ is therefore unique. For a fresh $N (0, s)$ variable $Z$ , [F3] implies that $Φ_{s} (Z)$ is uniform: for $0 < u < 1$ , the event $Φ_{s} (Z) \leq u$ is $Z \leq Φ_{s}^{- 1} (u)$ , of probability $u$ . Thus the inverse-CDF map has the asserted conditional law. If $Y = T_{j} (x^{'}, Z)$ and $U = Y - y_{0}$ , then $F_{v} (U) = Φ_{s} (Z)$ . Step 6.1 and strict monotonicity imply $∣ U - Z ∣ \leq ρ (U)$ whenever $U$ is moderate. Step 1.1 gives $∣ y_{0} ∣ / σ \leq C ϵ δ^{2} \leq C c δ$ , so $∣ x^{'} ∣ + ∣ Y ∣ \leq c s / τ$ implies the required moderate bounds on $U$ and $δ$ . The inequalities $2 δ ∣ U ∣ / σ \leq δ^{2} + U^{2} / s$ and $δ \leq 1 + δ^{2}$ now give $∣ Y - Z ∣ \leq C τ (1 + (∣ x^{'} ∣^{2} + Y^{2}) / s)$ .

8.1F3step 1.1step 6.1algebrachoose

The bound in Gaussian rather than target values follows by bracketing, not by substituting $Z$ for $Y$ in step 7.1. In standardized units fix a Gaussian coordinate $z$ and put $A = 1 + δ^{2} + z^{2}$ , $L = K_{1} ϵ A$ , with fixed large $K_{1}$ . If $ϵ δ$ and $ϵ ∣ z ∣$ are small enough, $z \pm L$ remain in the moderate region and the standardized version of $ρ (z \pm L)$ is at most $L / 2$ . Indeed $ρ (t) \leq C ϵ (1 + δ^{2} + t^{2})$ and $(z \pm L)^{2} \leq 2 z^{2} + 2 L^{2}$ ; the extra $C ϵ L^{2}$ is absorbed into $L$ because $ϵ^{2} A$ is small. Step 6.1 therefore gives $F_{g} (z - L) < Φ (z) < F_{g} (z + L)$ , so its inverse at $Φ (z)$ lies in $[z - L, z + L]$ . Adding the shift $y_{0}$ from step 1.1 and restoring units gives $∣ T_{j} (x^{'}, Z) - Z ∣ \leq C τ (1 + (∣ x^{'} ∣^{2} + Z^{2}) / s)$ under the stated Gaussian-value cutoff.

8.2F3F5F7step 2.1step 7.1algebra

The maps just constructed are jointly measurable, also under the additional parameter in the Statement. On each fixed row length the product of the bounded input-law kernels exists by [F5]. The Gaussian density is jointly Borel in its positive definite covariance and its argument: its determinant and inverse are continuous functions of the matrix entries by their polynomial and cofactor formulas. Consequently $p_{θ} (x) = \int q_{G_{θ}} (x - \sum_{i} t_{i}) ⨂_{i} K_{i} (θ, d t_{i})$ is jointly measurable by [F5], as are the prefix densities, their positive ratios and their CDF integrals. Lebesgue integration here uses its common exhaustion by bounded intervals. The conditional inverse can be written $\inf {r \in Q : H_{j, θ} (x^{'}, r) \geq u}$ ; by [F7] this equals the continuous strictly increasing inverse and is jointly measurable, since strict sublevel sets are countable unions of measurable sets indexed by $r \in Q$ . Compose with the continuous $Φ_{s} (z) = Φ (z / \sqrt{s})$ . Different finite row lengths are handled on their countable measurable disjoint union. This constructs the kernel versions explicitly at every prefix.

9.1F3F4step 7.1step 8.2algebra

Starting with independent Gaussian coordinates of variance $s$ , apply the conditional maps successively to coordinates $1, \dots, d$ . Conditional integration of the densities in step 2.1 shows inductively that the first $j$ constructed coordinates have density $p_{j}$ ; their joint density is the product of the successive density ratios, which telescopes to $p_{j}$ . The full map $R$ therefore has the law of $S$ . On $∣ R (g) ∣ \leq c s / τ$ , every coordinate and prefix meets the target-value cutoff in step 7.1 after reducing $c$ by a fixed factor. Summing at most four coordinate bounds gives $∣ R (g) - g ∣ \leq C τ (1 + ∣ R (g) ∣^{2} / s)$ . For $S = (P, D)$ apply only its last $a$ conditional maps with the specified prefix $P = p$ and fresh independent Gaussian coordinates; the same telescoping gives the conditional law of $D$ and the same coordinate estimates give $∣ T_{p} (e) - e ∣ \leq C τ (1 + (∣ p ∣^{2} + ∣ T_{p} (e) ∣^{2}) / s)$ . Separate bounds on the two norms imply the joint cutoff by reducing $c$ . The Gaussian reference has covariance $s I$ , so its parent and bridge coordinate blocks are independent by the product Gaussian density in [F3]. Joint measurability follows by composing the maps of step 8.2.

10.1F1F3step 8.2step 9.1algebra∎

For the scalar extension put $a = \sqrt{s / W}$ , $b = \sqrt{s / v}$ and $Q = (a P, b D)$ . Independence gives $Var (P) = W$ , $Cov (P, D) = V_{L} - w W = 0$ and $Var (D) = V_{L} V_{R} / W = v$ , hence $cov (Q) = s I_{2}$ . The stated ranges follow from $V_{L}, V_{R} \in [s / 2, s]$ ; the function $u v / (u + v)$ is increasing in either positive argument, giving $v \in [s / 4, s / 2]$ . Thus $a \in [1 / \sqrt{2}, 1]$ , $b \in [\sqrt{2}, 2]$ and $w \in [1 / 3, 2 / 3]$ . A bounded left increment $X$ contributes $(a X, b (1 - w) X)$ to $Q$ , whose norm is at most $2 B$ ; a bounded right increment contributes $(a X, - b w X)$ with the same bound. The joint Gaussian component is obtained from independent child Gaussians by the same linear map. Since their variances are at least $γ V_{L}, γ V_{R}$ , its covariance, by congruence, is at least $γ cov (Q) = γ s I_{2}$ . Therefore $Q$ satisfies the original hypotheses with bound $2 B$ and parameter $2 τ$ . Apply step 9.1 conditionally on $a P = a p$ , feeding $b e \sim N (0, s)$ , and divide the output by $b$ . The uniform bounds on $a, b$ convert its estimate and its cutoffs into the asserted scalar bridge bound after changing $c_{γ}, C_{γ}$ by fixed factors. Applying the scalar root construction to $a P$ and rescaling gives the root bound. Finally $D = L - w P$ rearranges to the two stated child formulas. Every coefficient is a continuous function of the positive variances, so the parameter measurability in step 8.2 is preserved. All conclusions follow.

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Harris geometric ergodicity admits an integrable total-variation envelope

Statement

Assume the Axiom of Choice. Let $P$ be a probability kernel on a measurable space $(X, X)$ with invariant probability $π$ . Assume it is $ϕ$ -irreducible (Phi-irreducibility for a general-state Markov kernel), aperiodic (Aperiodicity for a general-state Markov kernel), and Harris recurrent relative to its nonzero $σ$ -finite irreducibility measure $ϕ$ (Harris recurrence relative to an irreducibility measure). Suppose $P$ is geometrically ergodic in the sense of Geometric and uniform ergodicity for MCMC CLTs. Then there exist a measurable $M : X \to [0, \infty)$ with $π (M) < \infty$ and $r \in (0, 1)$ such that $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) r^{n} (x \in X, n \geq 1) .$

In fact, the same conclusion holds assuming only AC, invariance of $π$ , and the everywhere finite measurable common geometric-rate bound in Geometric and uniform ergodicity for MCMC CLTs, on the original arbitrary measurable space. The construction below derives the needed irreducibility on a restricted sigma-field directly from that bound.

Facts & Assumptions

Given: AC and all hypotheses in the Statement. Denote an everywhere finite measurable envelope in the geometric-ergodicity hypothesis by $M_{0}$ , with rate $r_{0} \in (0, 1)$ .

[A1]

The Axiom of Choice: AC is assumed for the small-set supplier and the canonical path-space and Markov suppliers below.

[F1]

Geometric and uniform ergodicity for MCMC CLTs: $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M_{0} (x) r_{0}^{n}$ for every $x$ and $n \geq 1$ , with $M_{0}$ finite and measurable.

[F2]

A kernel has a countably generated stable sub-sigma-field containing a given observable: a real measurable observable belongs to a countably generated sub-sigma-field on which the kernel restricts, with the same iterated transitions on restricted events and the restricted invariant law.

[F3]

A countably generated phi-irreducible chain has a positive-measure small set: a countably generated irreducible chain has a positive-irreducibility-measure small set $D$ , with a minorizing probability $ν_{0}$ satisfying $ν_{0} (D) > 0$ .

[F4]

Geometric small-set renewal convergence gives an integrable exponential hitting moment: a one-step small set of positive invariant mass, whose minorizing-law return probabilities converge geometrically, has a stationary-integrable exponential entrance-time moment.

[F5]

Canonical Markov chain on path space: under AC, the original kernel has canonical chains from every initial law.

[F6]

Discrete strong Markov property: bounded future-path functionals obey the Markov identity at stopping times.

[F7]

Phi-irreducibility for a general-state Markov kernel: irreducibility relative to a nonzero measure means that each positive-measure measurable set is accessible from every state at some positive time.

Proof

Given: The hypotheses in the Statement and the envelope $M_{0}, r_{0}$ in [F1], with AC [A1].

1.1A1F1F2F3F7given

Apply [F2] to $M_{0}$ to obtain a countably generated $G \subseteq X$ , the restricted kernel $P_{G}$ , and invariant probability $π_{G} = π ∣_{G}$ . If $B \in G$ has $π_{G} (B) > 0$ , [F1] implies $P^{n} (x, B) > 0$ for all sufficiently large $n$ , for each $x$ , because $M_{0} (x)$ is finite. Thus $P_{G}$ is $π_{G}$ -irreducible by [F7]. Applying [F3] on this countably generated space gives an $m \geq 1$ , a $D \in G$ with $π (D) > 0$ , a probability $ν_{0}$ on $G$ with $ν_{0} (D) > 0$ , and $ϵ_{0} > 0$ such that $P_{G}^{m} (x, \cdot) \geq ϵ_{0} ν_{0}$ on $D$ . No small-set assertion on the original sigma-field is being made.

2.1F1step 1.1chooseconstruct

Since $M_{0}$ is finite everywhere, $D \cap {M_{0} \leq K}$ increases to $D$ as the positive integer $K$ increases. Choose $K$ so $C = D \cap {M_{0} \leq K}$ has both $π (C) > 0$ and $ν_{0} (C) > 0$ . Choose a finite integer $L \geq K$ and put $a = ν_{0} (M_{0} \leq L) > 0$ , $ν (B) = ν_{0} (B \cap {M_{0} \leq L}) / a$ . Choose $0 < ϵ < \min (1, ϵ_{0} a)$ . For $Q = P_{G}^{m}$ this gives $Q (x, \cdot) \geq ϵ ν$ on $C$ , $ν (M_{0}) \leq L$ , and $M_{0} \leq K$ on $C$ .

3.1A1F1F2F4F5step 2.1

For $n \geq 1$ , the restricted-iteration identity in [F2] and [F1] give $∣ ν Q^{n} (C) - π (C) ∣ \leq \int M_{0} (x) r_{0}^{m n} ν (d x) \leq L (r_{0}^{m})^{n} .$ For $n = 0$ the absolute difference is at most $1$ . Therefore [F4] applies with constant $\max (1, L)$ and rate $r_{0}^{m}$ . It supplies $s > 1$ and a measurable $H (x) = E_{x} [s^{σ}]$ , where $σ = \inf {j \geq 0 : X_{j m} \in C}$ , such that $π (H) < \infty$ . Indeed the skeleton path restricted to $G$ under the full canonical $P$ -chain has the $Q$ -chain law: its finite-dimensional cylinder probabilities agree by the iterated-kernel identity in [F2]. Thus the same $H$ is the entrance-time moment for the original chain of [F5].

4.1A1F1F5F6step 2.1step 3.1choosealgebra

Choose $\max (r_{0}, s^{- 1 / m}) < r_{1} < 1$ , and set $S = m σ$ and $K_{1} = \max (1, K)$ . Fix $A \in X$ and $n \geq 1$ . On ${S \leq n}$ , decompose into the finitely many events ${S = j}$ and use [F6] to replace the conditional terminal indicator by $P^{n - j} (X_{j}, A)$ . Since $X_{j} \in C$ , [F1] bounds its difference from $π (A)$ by $K_{1} r_{0}^{n - j}$ ; when $j = n$ , use $∣ 1_{A} (X_{n}) - π (A) ∣ \leq 1 \leq K_{1}$ . On ${S > n}$ use the bound $1$ . Subtracting $π (A)$ on these two events gives $∣ P^{n} (x, A) - π (A) ∣ \leq P_{x} (S > n) + K_{1} E_{x} [r_{0}^{n - S}; S \leq n] \leq (1 + K_{1}) r_{1}^{n} H (x) .$ For the final inequality, $r_{1}^{- S} \leq s^{σ}$ , and on $S \leq n$ one has $r_{0}^{n - S} \leq r_{1}^{n} r_{1}^{- S}$ ; on $S > n$ , $1 \leq r_{1}^{n} r_{1}^{- S}$ . This argument is for each original measurable event $A$ , so taking its supremum controls the full original total variation and does not require measurability of a total-variation supremum.

5.1A1F1step 3.1step 4.1construct∎

The measurable set $N = {H = \infty}$ is $π$ -null by step 3.1. Define $M_{1} (x) = (1 + K_{1}) H (x)$ for $x \notin N$ and $M_{1} (x) = M_{0} (x)$ for $x \in N$ . It is measurable and finite everywhere, and $π (M_{1}) < \infty$ . Off $N$ , step 4.1 gives the desired bound with rate $r_{1}$ ; on $N$ , [F1] gives it because $r_{0} < r_{1}$ . Taking $M = M_{1}$ and $r = r_{1}$ proves the stronger conclusion from invariance and the everywhere geometric bound alone: the restricted irreducibility used in step 1.1 was derived there directly. The original additional Harris, irreducibility and aperiodicity hypotheses therefore give the same conclusion as a special case. AC was used precisely through [F3] and the path-space and Markov suppliers [F4], [F5] and [F6].

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Strong-mixing CLTs under bounded, moment, and logarithmic-moment conditions

Statement

Assume AC. Let $(Y_{k})_{k \geq 0}$ be a centered strictly stationary real sequence with $E [Y_{0}^{2}] < \infty$ , and define $S_{n} = \sum_{k = 1}^{n} Y_{k}$ . Each of the following is sufficient:

There is $B < \infty$ with $∣ Y_{0} ∣ \leq B$ almost surely and $\sum_{n \geq 1} α_{Y} (n) < \infty$ .
For some $δ > 0$ , $E ∣ Y_{0} ∣^{2 + δ} < \infty$ and $\sum_{n \geq 1} α_{Y} (n)^{δ / (2 + δ)} < \infty$ .
For some $a \in (0, 1)$ , $α_{Y} (n) = O (a^{n})$ and $E [Y_{0}^{2} \log^{+} ∣ Y_{0} ∣] < \infty$ , where $\log^{+} u = \max (0, \log u)$ for $u > 0$ and $0 \log^{+} 0 : = 0$ .

In each case the covariance series $σ_{Y}^{2} : = E [Y_{0}^{2}] + 2 \sum_{k = 1}^{\infty} Cov (Y_{0}, Y_{k})$ converges absolutely and is nonnegative, and $n^{- 1 / 2} S_{n} \Rightarrow N (0, σ_{Y}^{2}) .$ This includes the degenerate law $N (0, 0)$ when $σ_{Y}^{2} = 0$ , as in Zero covariance long-run variance gives a degenerate CLT under absolute summability. The coefficient $α_{Y}$ and strict stationarity have the meanings in Strong, uniform, and maximal-correlation mixing coefficients.

Facts & Assumptions

Given: AC, the centered strictly stationary square-integrable real sequence, and one of the three sufficient regimes.

[A1]

AC is assumed for the bounded CLT, normal-law characteristic function and characteristic-function convergence criterion. (The Axiom of Choice)

[F1]

A centered bounded stationary real sequence with summable strong-mixing coefficients has an absolutely summable covariance series and a CLT with its nonnegative covariance-sum variance, including variance zero. (Bounded stationary CLT under summable strong mixing)

[F2]

Absolute tail quantiles have the original absolute-value law under uniform Lebesgue measure; square-integrable variables from fields with event coefficient $a$ satisfy $∣ Cov (U, V) ∣ \leq 4 \int_{0}^{a} Q_{U} Q_{V}$ . (Strong-mixing covariance bound in terms of tail quantiles)

[F3]

A measurable scalar observation preserves strict stationarity and does not increase the strong-mixing coefficients. (A measurable observation does not increase stationary mixing coefficients)

[F4]

Dominated convergence applies on arbitrary measure spaces and therefore also to absolutely dominated covariance series. (Dominated convergence)

[F5]

Tonelli interchanges nonnegative product integrals, including integration against counting measure. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product)

[F6]

Holder's integral inequality holds for conjugate finite exponents on every measure space. (Holder's inequality for integrals, including the endpoint cases)

[F7]

Cauchy–Schwarz bounds absolute product expectations of square-integrable real variables. (Cauchy-Schwarz for random variables)

[F8]

Under AC the characteristic function of $N (0, s^{2})$ is $e^{- s^{2} t^{2} / 2}$ , including $s = 0$ . (Characteristic function of a normal law)

[F9]

Under AC, weak convergence to a specified probability law is equivalent to pointwise convergence of characteristic functions to that of the law. (Characteristic function criterion for weak convergence)

[F10]

Zero covariance-sum variance and absolute covariance summability give convergence of normalized sums in $L^{2}$ and in distribution to $N (0, 0)$ . (Zero covariance long-run variance gives a degenerate CLT under absolute summability)

[F11]

Variance and covariance are centered product expectations. (Moments, variance, and covariance on a probability space)

[F12]

The coefficient $α_{Y} (n)$ is the event supremum for the past and the future separated by $n$ . (Strong, uniform, and maximal-correlation mixing coefficients)

Proof

1.1A1F1F2F5givenalgebra

Write $Q = Q_{Y_{0}}$ for the absolute tail quantile in [F2], and put $w (u) = 1 + \sum_{r \geq 1} 1_{{u < α_{Y} (r)}}$ and $J = \int_{0}^{1} w (u) Q (u)^{2} d u$ . We first show in every stated regime that $J < \infty$ and $\sum_{r} α_{Y} (r) < \infty$ . In regime 1, $Q \leq B$ and [F5] give $J \leq B^{2} (1 + \sum_{r} α_{Y} (r)) < \infty$ . This regime also already follows directly from [F1].

2.1F2F5F6F12step 1.1givenalgebra

In regime 2 put $p = 2 + δ$ and $θ = 1 - 2 / p = δ / (2 + δ) \in (0, 1)$ . Holder's inequality [F6] with exponents $p / 2$ and $p / (p - 2)$ , followed by the quantile identity [F2], gives $\int_{0}^{α_{Y} (r)} Q (u)^{2} d u \leq α_{Y} (r)^{θ} {(\int_{0}^{1} Q (u)^{p} d u)}^{2 / p} = α_{Y} (r)^{θ} (E ∣ Y_{0} ∣^{p})^{2 / p} .$ Tonelli [F5] and the assumed summability yield $J < \infty$ . Also $0 \leq α_{Y} (r) \leq 1$ by its event definition, so $α_{Y} (r) \leq α_{Y} (r)^{θ}$ , proving unweighted summability.

2.2F2step 1.1givenalgebra

In regime 3, choose constants $C < \infty$ and $a \in (0, 1)$ so $α_{Y} (r) \leq C a^{r}$ outside a finite set of lags. Counting the positive integers satisfying $u < C a^{r}$ shows $w (u) \leq C_{1} + C_{2} \log (1 / u)$ for $0 < u < 1$ , with finite nonnegative constants $C_{1}, C_{2}$ ; also $\sum_{r} α_{Y} (r) < \infty$ . The elementary inequality $x v \leq x \log^{+} x + e^{v}$ for $x, v \geq 0$ follows for $x > 0$ by maximizing $x v - x \log x + x$ as a function of $x$ , whose maximum is $e^{v}$ ; $x = 0$ is immediate. Set $x = Q (u)^{2}$ and $v = \frac{1}{2} \log (1 / u)$ , then integrate and use [F2]: $\int_{0}^{1} Q (u)^{2} \log (1 / u) d u \leq 2 \int_{0}^{1} Q (u)^{2} \log^{+} (Q (u)^{2}) d u + 2 \int_{0}^{1} u^{- 1 / 2} d u = 4 E [Y_{0}^{2} \log^{+} ∣ Y_{0} ∣] + 4 < \infty .$ Together with the assumed second moment this proves $J < \infty$ . Thus the two properties announced in step 1.1 hold in all three regimes, without imposing a power moment in regime 3.

3.1F2F4F5F11F12step 1.1step 2.1step 2.2algebra

Let $γ_{r} = Cov (Y_{0}, Y_{r})$ . By [F2], [F12] and [F5], $\sum_{r \geq 1} ∣ γ_{r} ∣ \leq 4 \sum_{r \geq 1} \int_{0}^{α_{Y} (r)} Q (u)^{2} d u = 4 \int_{0}^{1} (w (u) - 1) Q (u)^{2} d u < \infty .$ Set $D_{Y} = γ_{0} + 2 \sum_{r \geq 1} ∣ γ_{r} ∣$ . Expanding a finite centered sum over $I \subseteq N_{0}$ and counting at most $∣ I ∣$ pairs at each positive lag gives $E (\sum_{i \in I} Y_{i})^{2} \leq D_{Y} ∣ I ∣$ . For consecutive sums the exact expression is $Var (S_{n}) / n = γ_{0} + 2 \sum_{r = 1}^{n - 1} (1 - r / n) γ_{r}$ . The summable majorant and [F4] prove convergence to the stated $σ_{Y}^{2}$ , which is nonnegative as a limit of nonnegative variances.

3.2A1F1F3step 1.1step 2.1step 2.2construct

For an integer $M \geq 1$ put $c_{M} (y) = \max (- M, \min (y, M))$ , $Z_{k}^{(M)} = c_{M} (Y_{k}) - E c_{M} (Y_{0})$ , and $H_{M} (y) = y - c_{M} (y)$ . Then $R_{k}^{(M)} = Y_{k} - Z_{k}^{(M)} = H_{M} (Y_{k}) - E H_{M} (Y_{0})$ . Both processes are strictly stationary by [F3], and $Z^{(M)}$ is centered and bounded by $2 M$ . Its coefficients are bounded by $α_{Y}$ , whose sum is finite by steps 1.1, 2.1, 2.2. Thus [F1] gives an absolutely convergent covariance series with nonnegative sum $σ_{M}^{2}$ , normalized variance convergence to $σ_{M}^{2}$ , and $n^{- 1 / 2} \sum_{k = 1}^{n} Z_{k}^{(M)} \Rightarrow N (0, σ_{M}^{2})$ .

4.1F2F4F5F11F12step 1.1step 2.1step 2.2step 3.1step 3.2algebra

Put $t_{M} = P (∣ Y_{0} ∣ > M)$ and let $Q_{H_{M}}$ be the absolute quantile of $H_{M} (Y_{0})$ . Since $∣ H_{M} (Y_{0}) ∣ = (∣ Y_{0} ∣ - M)_{+} \leq ∣ Y_{0} ∣$ , monotonicity of tail probabilities gives $Q_{H_{M}} (u) \leq Q (u)$ , while $Q_{H_{M}} (u) = 0$ for $u \geq t_{M}$ by its defining infimum. The centering in $R^{(M)}$ does not change any covariance, and $Var (H_{M} (Y_{0})) \leq E H_{M} (Y_{0})^{2}$ . Applying [F2] to the raw, uncentered $H_{M} (Y_{k})$ , followed by [F5], therefore gives $D_{R^{(M)}} : = E (R_{0}^{(M)})^{2} + 2 \sum_{r \geq 1} ∣ Cov (R_{0}^{(M)}, R_{r}^{(M)}) ∣ \leq 8 \int_{0}^{t_{M}} w (u) Q (u)^{2} d u = : e_{M}^{2} .$ Here the integrand is integrable by steps 1.1, 2.1, 2.2, and $t_{M} \to 0$ because $Y_{0}$ is finite almost surely; hence [F4] makes $e_{M} \to 0$ . The same finite covariance expansion as in step 3.1 now gives $\sup_{n} ∥ n^{- 1 / 2} \sum_{k = 1}^{n} R_{k}^{(M)} ∥_{2} \leq e_{M}$ . This bound uses the raw remainder quantile, without asserting that the centered remainder has that quantile.

5.1F7step 3.1step 3.2step 4.1algebra

The reverse triangle inequality for $L^{2}$ norms gives $∣ ∥ S_{n} / \sqrt{n} ∥_{2} - ∥ n^{- 1 / 2} \sum_{k = 1}^{n} Z_{k}^{(M)} ∥_{2} ∣ \leq e_{M}$ by step 4.1. This norm inequality follows by expanding a square and applying [F7] to its mixed term. Using the variance limits in steps 3.1 and 3.2 yields $∣ σ_{Y} - σ_{M} ∣ \leq e_{M}$ , where the square roots are nonnegative. Therefore $σ_{M} \to σ_{Y}$ .

6.1A1F7F8F9F10step 3.1step 3.2step 4.1step 5.1∎

For fixed real $t$ , $∣ e^{i x} - e^{i y} ∣ \leq ∣ x - y ∣$ , obtained by integrating the unit-modulus derivative along the real interval, and [F7] give $∣ E e^{i t S_{n} / \sqrt{n}} - E e^{i t \sum_{k = 1}^{n} Z_{k}^{(M)} / \sqrt{n}} ∣ \leq ∣ t ∣ e_{M}$ . By the CLT in step 3.2 and [F9], the second characteristic function tends to $e^{- t^{2} σ_{M}^{2} / 2}$ , identified by [F8]. Consequently ${lim sup}_{n \to \infty} ∣ E e^{i t S_{n} / \sqrt{n}} - e^{- t^{2} σ_{Y}^{2} / 2} ∣ \leq ∣ t ∣ e_{M} + ∣ e^{- t^{2} σ_{M}^{2} / 2} - e^{- t^{2} σ_{Y}^{2} / 2} ∣ .$ Let $M \to \infty$ and apply step 5.1. Under [A1], [F8]–[F9] prove $S_{n} / \sqrt{n} \Rightarrow N (0, σ_{Y}^{2})$ . The case $σ_{Y}^{2} = 0$ is also independently supplied by [F10] using absolute covariance summability from step 3.1. All three regimes and the degenerate case follow.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

A uniform-mixing CLT for square-integrable stationary sequences

Statement

Assume AC. Let $(Y_{k})_{k \geq 0}$ be a centered strictly stationary real sequence with $E [Y_{0}^{2}] < \infty$ . If $\sum_{n = 1}^{\infty} \sqrt{ϕ_{Y} (n)} < \infty,$ then the covariance series $σ_{Y}^{2} : = E [Y_{0}^{2}] + 2 \sum_{k = 1}^{\infty} Cov (Y_{0}, Y_{k})$ converges absolutely and is nonnegative. If $σ_{Y}^{2} > 0$ , then for $S_{n} = \sum_{k = 1}^{n} Y_{k}$ , $n^{- 1 / 2} S_{n} \Rightarrow N (0, σ_{Y}^{2}) .$ If $σ_{Y}^{2} = 0$ , Zero covariance long-run variance gives a degenerate CLT under absolute summability gives the degenerate normal limit.

Facts & Assumptions

Given: AC, the centered strictly stationary real sequence $Y$ , finite second moment, and $\sum_{n \geq 1} \sqrt{ϕ_{Y} (n)} < \infty$ .

[A1]

AC is inherited through the rho-mixing CLT. (The Axiom of Choice)

[F1]

The coefficient $ϕ_{Y} (n)$ bounds $∣ P (B ∣ A) - P (B) ∣$ for a positive-probability past event $A$ and an event $B$ in the separated future; $ρ_{Y} (n)$ is the supremum of their square-integrable variable correlations. (Strong, uniform, and maximal-correlation mixing coefficients)

[F2]

Real square-integrable random variables satisfy Cauchy–Schwarz. (Cauchy-Schwarz for random variables)

[F3]

Simple functions are dense in real $L^{2}$ on a probability space. (Simple functions with finite-measure support are dense in $L^{p} (μ)$ for $1 \leq p < \infty$ )

[F4]

For a centered stationary real sequence of finite second moment, $\sum_{n \geq 1} ρ_{Y} (n) < \infty$ gives absolute convergence and nonnegativity of the covariance variance, and the normal CLT with that variance, including zero. (A rho-mixing CLT for square-integrable stationary sequences)

[F5]

Variance and covariance are centered second and product moments. (Moments, variance, and covariance on a probability space)

Proof

1.1F1givenconstructalgebra

Fix a gap $n \geq 1$ and a split $k \geq 0$ , put $A = σ (Y_{0}, \dots, Y_{k})$ and $B = σ (Y_{k + n}, Y_{k + n + 1}, \dots)$ , and put $ϕ = ϕ_{Y} (n)$ . Take finite partitions $(A_{i})$ in $A$ and $(B_{j})$ in $B$ . Write $a_{i} = P (A_{i})$ , $b_{j} = P (B_{j})$ , $p_{i j} = P (A_{i} \cap B_{j})$ , and $d_{i j} = p_{i j} - a_{i} b_{j}$ . For each fixed $i$ the $d_{i j}$ sum to zero. If $a_{i} > 0$ , their positive sum is $P (A_{i} \cap B) - a_{i} P (B) \leq ϕ a_{i}$ for $B = ⋃_{j : d_{i j} > 0} B_{j}$ , by [F1]. Hence $\sum_{j} ∣ d_{i j} ∣ \leq 2 ϕ a_{i}$ ; when $a_{i} = 0$ , every $d_{i j} = 0$ and the same bound holds. For each $j$ , $∣ p_{i j} - a_{i} b_{j} ∣ \leq p_{i j} + a_{i} b_{j}$ gives $\sum_{i} ∣ d_{i j} ∣ \leq 2 b_{j}$ .

2.1F2F5step 1.1algebra

For simple $U = \sum_{i} u_{i} 1_{A_{i}}$ and $V = \sum_{j} v_{j} 1_{B_{j}}$ , [F5] and finite Cauchy–Schwarz with weights $∣ d_{i j} ∣$ give $∣ Cov (U, V) ∣ = ∣ \sum_{i, j} d_{i j} u_{i} v_{j} ∣ \leq (\sum_{i, j} ∣ d_{i j} ∣ u_{i}^{2})^{1 / 2} (\sum_{i, j} ∣ d_{i j} ∣ v_{j}^{2})^{1 / 2} \leq 2 \sqrt{ϕ} ∥ U ∥_{2} ∥ V ∥_{2}$ , using step 1.1's row and column bounds. The finite Cauchy–Schwarz inequality here is also the probability-space inequality [F2] on a finite set, after normalizing the total weight; if all weights vanish the conclusion is immediate.

3.1F1F2F3F5step 2.1

For arbitrary real square-integrable $A$ -measurable $U$ and $B$ -measurable $V$ , choose simple approximants in their respective $L^{2}$ spaces using [F3]. Their means, norms and product expectations converge by [F2]; therefore step 2.1's inequality passes to the limit. Apply it to $U - E U$ and $V - E V$ to get $∣ Cov (U, V) ∣ \leq 2 \sqrt{ϕ} \sqrt{Var (U) Var (V)}$ . Taking the supremum over positive-variance variables and then splits in [F1] proves $ρ_{Y} (n) \leq 2 \sqrt{ϕ_{Y} (n)}$ . No reversal of the conditioning direction of $ϕ$ has been used.

4.1A1F4step 3.1given∎

Summation of step 3.1 gives $\sum_{n \geq 1} ρ_{Y} (n) \leq 2 \sum_{n \geq 1} \sqrt{ϕ_{Y} (n)} < \infty$ . The given stationarity, centering, and second-moment assumptions are exactly those of [F4], so that theorem proves absolute convergence and nonnegativity of the displayed covariance series and $S_{n} / \sqrt{n} \Rightarrow N (0, σ_{Y}^{2})$ for positive or zero $σ_{Y}^{2}$ . Thus every claimed conclusion holds.

Source notes

The phi-to-rho bound is proved directly by finite partitions and $L^{2}$ approximation. The normal limit is supplied by the locally proved rho-mixing CLT, rather than an external uniform-mixing limit theorem.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Bounded independent vector rows admit a logarithmic cubed Gaussian coupling

Statement

Assume AC. For every $A > 0$ there is a finite constant $C_{A}$ , independent of all row laws and lengths, such that the following holds. Let $1 \leq d \leq 2$ , $N \geq 1$ , $B \geq 1$ , and let $X_{1}, \dots, X_{N}$ be independent random vectors in $R^{d}$ with $E X_{i} = 0, cov (X_{i}) = I_{d}, ∣ X_{i} ∣ \leq B almost surely .$ Their distributions need not be identical. There is a coupling with independent $G_{1}, \dots, G_{N} \sim N_{d} (0, I_{d})$ such that $\Pr ⁣ (\max_{1 \leq k \leq N} ∣ \sum_{i = 1}^{k} (X_{i} - G_{i}) ∣ > C_{A} B \log^{3} (e N)) \leq N^{- A} .$ The coupling can be realized on an extension preserving an already realized raw row as its original marginal. Independence between the raw and Gaussian rows is not asserted.

Facts & Assumptions

Given: AC, $A, d, N, B$ , and the independent row with the displayed moment, covariance and almost-sure bounds. All norms are Euclidean.

[A1]

AC is assumed as in The Axiom of Choice.

[F1]

A sum of independent centered bounded vectors of norm at most $b$ and an independent Gaussian, with total covariance $s I_{j}$ and Gaussian covariance at least $γ s I_{j}$ , has the uniform analytic cumulant and tilted Fourier bounds of Gaussian anchored bounded sums have uniform cumulant and Fourier bounds. The bounds persist under row isometries and coordinate prefixes in dimensions at most four.

[F2]

The smooth Gaussian-anchored conditional quantile supplier gives measurable root and conditional bridge transports with their prescribed marginal laws. For fixed positive Gaussian covariance fraction, total covariance $s I$ , bounded-summand scale $τ = C b$ , and $s \geq C τ^{2}$ , their Gaussian-to-target errors are at most $C τ (1 + ∣ u ∣^{2} / s)$ at a root and $C τ (1 + (∣ u ∣^{2} + ∣ v ∣^{2}) / s)$ at a split with target parent $u$ and target difference $v$ , whenever these target norms are at most $c s / τ$ . Gaussian bridge coordinates are independent of the Gaussian parent. Conditional quantile couplings for Gaussian anchored sums

[F3]

Finite-dimensional Euclidean spaces are standard Borel, and a joint probability law on two such spaces has a probability-kernel disintegration given either marginal. Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel, Disintegration of a joint law on standard borel spaces

[F4]

A probability-kernel extension depending only on one part of an independent pair preserves independence of that part together with the newly sampled variable from the other part. Conditional independence under kernel extension and conditioning

[F5]

Markov's inequality applies to nonnegative random variables, and the probability of a finite union is at most the sum of its probabilities. Markov's inequality for random variables, The finite union bound

[F6]

Gaussian characteristic functions identify their covariance; jointly Gaussian subvectors of zero cross covariance are independent, by factorization of that characteristic function. Characteristic function of a multivariate normal law

[F7]

Increasing nonnegative integrands may be passed through the integral. Monotone convergence for the integral

[F8]

A nondegenerate multivariate normal law has its stated Gaussian density. Completing the square in that density gives the standard-normal mgf $E e^{h Z} = e^{h^{2} / 2}$ . A positive-definite multivariate normal law has its Gaussian density

[F9]

The Euclidean norm obeys the triangle inequality and is convex. Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

Proof

1.1givenA1F3F7constructchoose

It suffices to construct a coupling of row laws: by [F3] its joint law can be disintegrated given the raw row, and the measure $P (d ω) K (X (ω), d g)$ extends any original realization with that coupling law. This is a probability measure: for disjoint product-measurable events, countable additivity follows by monotone convergence inside the kernel integral. The original marginal remains $P$ because the kernel has total mass one. Pad the row to dyadic length $n = 2^{⌈ \log_{2} N ⌉}$ , with $n = 1$ if $N = 1$ , by independent extra entries of the law of $X_{1}$ . It suffices to control its first $N$ entries. Set $L = a \log (e n)$ , where the constant $a$ will depend only on $A$ .

1.2givenF5F8algebra

Every independent row of centered entries, each either one of the raw laws or $N_{d} (0, I_{d})$ , obeys a common Bernstein estimate. For a raw coordinate $Z$ , $E ∣ Z ∣^{r} \leq B^{r - 2}$ for $r \geq 2$ , so for $B ∣ h ∣ < 1$ exponential-series expansion gives $\log E e^{h Z} \leq h^{2} / [2 (1 - B ∣ h ∣)]$ ; the same inequality holds for a standard normal because its log mgf is $h^{2} / 2$ . For any block of $s$ entries, multiply the mgfs and apply [F5] to their exponential, using $h = u / (s + B u)$ and both signs, and then the coordinate union bound. Thus for $t \geq 1$ , $\Pr (∣ \sum_{i \in I} Z_{i} ∣ > C (\sqrt{s t} + B t)) \leq 4 e^{- t} .$ The constant is uniform in $d \leq 2$ and in the individual row laws.

1.3F1F2givenconstructalgebra

Consider a compressed dyadic row of length $m$ , whose odd entries are Gaussian and whose even entries have their prescribed raw laws. For every aligned dyadic interval of length $s \geq 4$ , both halves contain half Gaussian slots. Its sum has covariance $s I_{d}$ and Gaussian covariance $(s / 2) I_{d}$ . Its joint normalized parent and difference $((U_{L} + U_{R}) / \sqrt{2}, (U_{L} - U_{R}) / \sqrt{2})$ has covariance $(s / 2) I_{2 d}$ and Gaussian covariance $(s / 4) I_{2 d}$ . Each bounded contribution to this joint vector has norm at most $B$ . Hence [F1] applies with fixed Gaussian fraction $1 / 2$ , also to all coordinate prefixes, and [F2] supplies root and bridge maps. Rescaling the normalized pair only changes universal constants. No identical-distribution assumption was used: each bounded contribution retains its own original index law.

2.1F2step 1.3constructchoosealgebra

Put $τ = C B$ , with a fixed $C$ covering [F2], and choose a fixed large $H$ . Let $m_{0}$ be the least dyadic integer at least $H τ^{2} L$ . If $m \geq 2 m_{0}$ , choose an independent Gaussian root $V_{r o o t} \sim N_{d} (0, m I_{d})$ and independent Gaussian split primitives $Z_{I} \sim N_{d} (0, ∣ I ∣ I_{d})$ for all dyadic $I$ of length greater than $m_{0}$ . The root map constructs $U_{r o o t}$ from $V_{r o o t}$ . Descending the tree, with $U_{I}, V_{I}$ present, let $D_{I}$ be the conditional bridge map of $Z_{I}$ given target parent $U_{I}$ , and put $U_{I_{L}} = (U_{I} + D_{I}) / 2, U_{I_{R}} = (U_{I} - D_{I}) / 2, V_{I_{L}} = (V_{I} + Z_{I}) / 2, V_{I_{R}} = (V_{I} - Z_{I}) / 2.$ The new primitive is independent of the exposed parents. Its Gaussian conditional law therefore does not depend on $U_{I} - V_{I}$ . The target split has exactly the prescribed conditional law given $U_{I}$ by [F2].

3.1F2F3F6step 2.1constructalgebra

At terminal blocks sample the entries of the mixed independent row by their regular conditional law given their sum $U_{I}$ , with product kernels over distinct terminal blocks. Recover Gaussian entries given sum $V_{I}$ by fresh independent standard normals $h_{i}$ and the formula $g_{i} = V_{I} / m_{0} + h_{i} - m_{0}^{- 1} \sum_{j \in I} h_{j}$ . For the mixed row, independence of the original disjoint blocks implies that their conditional kernels given their sums factor: check the identity on products of block events by the disintegration formula and extend to all events by uniqueness of probability measures. Induction over the binary splits and then these terminal kernels recovers exactly the mixed row product law. For Gaussians, $(V_{I} + Z_{I}) / 2$ and $(V_{I} - Z_{I}) / 2$ have zero cross covariance and covariance $(∣ I ∣ / 2) I_{d}$ ; [F6] and induction give the Gaussian product law at each level and after terminal reconstruction. Thus the final two marginal rows have independent entries, despite their coupling.

4.1F2F5step 1.2step 2.1step 3.1choosealgebra

Apply step 1.2 with $t = L$ to every dyadic mixed sum and to every prefix within a terminal block in either completed marginal row. There are at most $4 m$ such vectors; their unconditional product marginal laws were proved in step 3.1. The union bound gives a good event of failure at most $C m e^{- c L}$ on which each mixed sum satisfies $∣ U_{I} ∣ \leq C (\sqrt{∣ I ∣ L} + τ L)$ and each terminal residual is at most $C (\sqrt{m_{0} L} + τ L)$ . At an active split of size $s \geq 2 m_{0}$ , $∣ D_{I} ∣ \leq ∣ U_{I_{L}} ∣ + ∣ U_{I_{R}} ∣ \leq C (\sqrt{s L} + τ L)$ . Choose $H$ large enough that both $∣ U_{I} ∣, ∣ D_{I} ∣ \leq c s / τ$ , using the bounds $C / \sqrt{H} + C / H$ for their ratios to $s / τ$ . Their normalized squares are at most $C L$ . The same holds at the root. Consequently [F2] bounds the root and every split discrepancy by $C τ L$ .

5.1step 1.2step 4.1F5F9algebra

At terminal boundaries write $E (k) = \sum_{i \leq k} (Z_{i} - g_{i})$ . Its endpoint values are zero and $U_{r o o t} - V_{r o o t}$ . At the midpoint $h$ of any active interval $[a, b]$ , the split formulas give $E (h) = (E (a) + E (b)) / 2 + (D_{I} - Z_{I}) / 2$ . Convexity of the Euclidean norm shows that adding one level increases the maximal boundary error by at most half the maximal split discrepancy at that level. There are at most $\log_{2} m$ levels. Adding the terminal residuals from step 4.1, and using $m_{0} < 2 H τ^{2} L$ , proves that one round's maximum prefix discrepancy is at most $C τ L \log (e n)$ except on probability $C m e^{- c L}$ . If $m < 2 m_{0}$ , use instead an independent coupling of the full mixed and Gaussian rows; step 1.2 and a union bound over all their initial prefixes bound their discrepancy by $C (\sqrt{m L} + τ L) \leq C τ L$ , with the same failure bound. This also covers $m = 1$ and arbitrarily large $B$ .

6.1F3F4step 1.3step 5.1construct

Start with an independent Gaussian row of length $n$ . Apply the round just proved to that row, putting its designated raw law in every even slot and retaining Gaussian odd slots. This is possible while preserving the existing row: disintegrate the coupling given its Gaussian marginal using [F3] and extend by that kernel. Freeze the newly raw entries. The output mixed marginal is a product law, so the retained Gaussian subsequence is independent of all frozen variables. Repeat on that subsequence using a kernel depending only on it; [F4] preserves independence from the frozen complement. At each stage every newly raw coordinate has the law originally designated for that particular original index. If the active length is less than $2 m_{0}$ , replace all its remaining slots by independent raw entries instead; step 5.1's terminal bound applies to the raw and Gaussian marginal prefix sums. After at most $⌈ \log_{2} n ⌉ + 1$ stages the final full row has exactly the prescribed product of the possibly different raw laws.

7.1F5F9step 1.1step 5.1step 6.1choosealgebra∎

A prefix of the original row meets each active subsequence in an initial prefix, so triangle inequality bounds its total change by the sum of the stage maximum prefix errors. Each is at most $C τ L \log (e n)$ and there are at most $C \log (e n)$ stages. The total error is therefore at most $C_{A} B \log^{3} (e n)$ . Use the same ambient $L = a \log (e n)$ in every stage; since each active length is at most $n$ , the total failure probability is at most $C n \log (e n) e^{- c a \log (e n)}$ . Choose $a$ large enough, depending only on $A$ , that this is at most $N^{- A}$ , including the finitely small lengths by increasing $a$ once. As $n < 2 N$ for $N > 1$ , $\log (e n) \leq C \log (e N)$ . Restricting to the first $N$ coordinates and using step 1.1 proves the claimed estimate and the assertion about preserving an existing raw realization.

Source notes

Zaitsev's cited Sections 5–8 give the dyadic conditional-quantile and successive-replacement architecture. The present proof uses half Gaussian anchors and a stopped tree, allowing one extra logarithmic factor and avoiding third-moment matching. It proves only the displayed polynomial-tail logarithmic cubed bound, with constants uniform over the bounded nonidentical row laws; it does not claim the source's sharper exponential bound.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Geometric Markov chains admit delayed adjacent-iid split tours with spare moments

Statement

Assume AC. Let $P$ be a probability kernel on a nonempty standard-Borel space $(S, S)$ , with invariant probability $π$ , finite measurable $M : S \to [0, \infty)$ and $0 < r < 1$ such that $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) r^{n} (x \in S, n \geq 1) .$ Let $f : S \to R$ be finite measurable, with $π (f) = 0$ and $π (∣ f ∣^{v}) < \infty$ for some $v > 1$ . For every initial law, on an extension preserving the given chain there are an integer $m \geq 1$ , an almost surely finite delay $D \in m N$ , a standard-Borel seed space $E$ , iid seeds $(U_{j})_{j \geq 0}$ , and measurable finite-word and length functions $W : E^{2} \to ⨆_{k \geq 1} S^{k}$ and $L : E \to m N$ with the following properties.

Put $T_{0} = D$ and $T_{j + 1} = T_{j} + L (U_{j})$ . Almost surely, simultaneously for all $j \geq 0$ , $(X_{T_{j}}, \dots, X_{T_{j + 1} - 1}) = W (U_{j}, U_{j + 1}) .$ Write $L_{j} = L (U_{j})$ , $Z_{j} = \sum_{t = T_{j}}^{T_{j + 1} - 1} f (X_{t})$ , and $H_{j} = \sum_{t = T_{j}}^{T_{j + 1} - 1} ∣ f (X_{t}) ∣$ . Then $Z_{j} = z (U_{j}, U_{j + 1})$ and $H_{j} = h (U_{j}, U_{j + 1})$ for measurable functions $z, h$ , and $E e^{a L_{0}} < \infty for some a > 0, 0 < μ_{L} : = E L_{0} < \infty,$ $E Z_{0} = 0, E H_{0}^{q} < \infty (1 \leq q < v) .$ For every nonnegative measurable $b : S \to [0, \infty]$ the full-tour occupation identity is $E \sum_{t = T_{0}}^{T_{1} - 1} b (X_{t}) = μ_{L} π (b) .$ For each specified initial law, the same $m$ , iid seed law, $W, L$ and post-delay tour law can be used also for the invariant initial law $π$ ; only the finite initial segment and coupling change.

In particular $(L_{j} - μ_{L}, Z_{j})$ is an adjacent-iid centered vector sequence, while the lengths themselves are iid. Full tour words or rewards are not asserted to be iid. No moment of $D$ or of the initial finite prefix is required, and the construction uses a useful multistep split chosen from $P$ rather than requiring any supplied one-step minorization to have positive stationary mass.

Facts & Assumptions

Given: AC and the standard-Borel chain, geometric bound, invariant law, and finite centered observable in the Statement.

[A1]

The Axiom of Choice: AC supplies the inherited small-set, path-law, coding, product and disintegration conventions.

[F1]

Phi-irreducibility for a general-state Markov kernel and A countably generated phi-irreducible chain has a positive-measure small set: on a countably generated irreducible state space there is a positive-measure $m$ -small set, with a minorizing probability giving that set positive mass.

[F2]

Geometric small-set renewal convergence gives an integrable exponential hitting moment: under its scalar geometric-return-probability hypothesis the explicitly specified split atom has an exponential positive return moment and finite mean.

[F3]

Disintegration of a joint law on standard borel spaces and Finite products of standard Borel spaces are standard Borel: finite standard-Borel joint laws have measurable conditional probability kernels.

[F4]

Standard borel spaces admit bimeasurable real codings, Standard Borel spaces, and Under the Axiom of Countable Choice, countable products and $G_{δ}$ subspaces of Polish spaces are Polish: standard-Borel spaces have Borel real codings; countable sequences of such coordinates and finite words are standard Borel through Borel subsets of the Polish cube.

[F5]

Canonical Markov chain on path space, Assuming countable and dependent choice, countable products of arbitrary probability spaces, and Ionescu-Tulcea construction of a Markov chain: canonical kernel paths, iid products and extensions by specified kernels exist under AC.

[F6]

Measurability of integration against a kernel, Monotone convergence for the integral, and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: the kernel expectations and nonnegative sums below are measurable, and increasing limits pass through them.

[F7]

Holder's inequality for integrals, including the endpoint cases and Minkowski's inequality for integrals, including $p = \infty$ : the indicated Holder and finite-sum $L^{q}$ bounds hold for $q \geq 1$ .

[F8]

Invariant and stationary distribution for a Markov kernel and Iterated transition kernels: $π P = π$ and iterated transitions are successive kernel integrations; consequently $π P^{k} = π$ for every $k$ by induction.

[F9]

Dynkin's pi-lambda theorem: equality of probabilities or kernel identities extends from a countable generating algebra to all measurable events.

Proof

Given: The hypotheses in the Statement and AC [A1].

1.1A1F1F4givenchooseconstruct

For every $π$ -positive measurable set $A$ , the geometric bound implies $P^{n} (x, A) > 0$ for all sufficiently large $n$ , for each $x$ . Thus $P$ is $π$ -irreducible. By [F4], a standard-Borel space has a countable generating algebra, so [F1] gives $m \geq 1$ , $D_{0}$ with $π (D_{0}) > 0$ , a probability $ν_{0}$ with $ν_{0} (D_{0}) > 0$ , and $ϵ_{0} > 0$ such that $P^{m} (x, \cdot) \geq ϵ_{0} ν_{0}$ on $D_{0}$ . Choose an integer $K$ so $C = D_{0} \cap {M \leq K}$ is positive under both $π$ and $ν_{0}$ , and restrict $ν_{0}$ to ${M \leq L}$ for a finite integer $L \geq K$ , then normalize. Decrease the minorization constant to $0 < ϵ < 1$ . With $Q = P^{m}$ , the resulting probability $ν$ satisfies $Q (x, \cdot) \geq ϵ ν$ on $C$ , $ν (M) \leq L$ and $π (C) > 0$ .

1.2A1F3F4F6construct

The bridge kernel has a jointly measurable uniform sampler. Code its standard-Borel word space into a Borel subset of $[0, 1]$ by [F4]. Its distribution-function evaluations at rational cuts are measurable. The generalized inverse, defined by an infimum over those cuts, is jointly measurable in the endpoints and a uniform variable and has the coded probability law. Its output lies in the coded image almost surely for each endpoint; use a fixed valid word on the Borel complement and decode. For completeness, the word and sequence spaces here are standard Borel: code $S$ into a Borel subset $A \subseteq [0, 1]$ ; $A^{N}$ is Borel in the Polish cube by coordinate tests, and finite words are the Borel union of length-marked sequences with fixed trailing coordinates. The extra uniform sequences have the same property.

2.1F2F8step 1.1given

Put $p = ϵ π (C) > 0$ . Invariance [F8] gives $π \geq p ν$ . The geometric bound gives $∣ ν Q^{n} (C) - π (C) ∣ \leq L (r^{m})^{n}$ for $n \geq 1$ ; for $n = 0$ use $1$ . Thus [F2] applies to this exact split. Its atom $α = C \times {1}$ has a positive return time $τ$ with $E b^{τ} < \infty$ for some $b > 1$ . The transition out of $α$ draws the next skeleton state from $ν$ independently of the previous state, and assigns its fresh flag with success probability $ϵ 1_{C}$ .

3.1F5F6F8step 1.1step 2.1givenalgebra

Define the substochastic failure kernel $K_{0} (x, d y) = Q (x, d y) - ϵ 1_{C} (x) ν (d y)$ , and let $a (x)$ be the probability of never obtaining a successful split coin, starting with a fresh coin at $x$ . This is measurable by [F5] and [F6], lies in $[0, 1]$ , and satisfies $a = K_{0} a = Q a - ϵ 1_{C} ν (a)$ . Integrate against $π$ to get $p ν (a) = 0$ , hence $ν (a) = 0$ . Therefore $a = Q a$ at every state and $a = Q^{n} a$ for all $n$ . Convergence of $Q^{n} (x, \cdot)$ to $π$ implies $a (x) = π (a)$ for every $x$ , since for a measurable $[0, 1]$ -valued function its expectation difference is bounded by total variation, by integrating level-set indicators. Integrating this constant against $ν$ makes it zero. Thus every state and every initial law has a finite first success; its post-transition calendar time is an almost surely finite $D \in m N$ .

4.1F5F6step 2.1step 3.1construct

A skeleton core starts from the $ν$ endpoint just after a success and includes the successive states through the next successful coin, excluding that successful transition's endpoint. A core of length $ℓ$ has probability measure $ν (d x_{0}) \prod_{i = 0}^{ℓ - 2} K_{0} (x_{i}, d x_{i + 1}) ϵ 1_{C} (x_{ℓ - 1}) .$ The sum over $ℓ \geq 1$ has mass one by step 3.1. Its length has the law of $τ$ in step 2.1: starting in the split atom, one transition draws its initial $ν$ state, and the next atom visit occurs at precisely that core's inclusive coin count. Hence its length has an exponential moment. Taking iid copies of this core measure supplies all post-success cores; the first state of the next copy supplies the independent $ν$ endpoint of the preceding successful transition. Multiplying these core kernels reconstructs the split skeleton law. The analogous initial core starts from the given initial law and ends at its first success, before the endpoint drawn from the first iid core.

4.2A1F3F6F8F9step 2.1step 3.1construct

To restore the intermediate states, choose $ρ = \frac{1}{2} π + \frac{1}{2} \sum_{k \geq 0} 2^{- k - 1} μ P^{k}$ for the given initial law $μ$ . Then $π, μ ≪ ρ$ and $ρ P ≪ ρ$ , by the null-set calculation used directly on this displayed sum. Disintegrate the intermediate word $(X_{1}, \dots, X_{m - 1})$ conditional on $(X_{0}, X_{m})$ under initial law $ρ$ , using [F3]; for $m = 1$ take the unique empty word. Denote the kernel by $B (x, y; d w)$ . Its bridge identity holds for $ρ$ -almost every starting point, simultaneously as a probability identity by [F9]. Every starting distribution encountered in a split skeleton is $ρ$ -absolutely continuous: $K_{0} \leq Q$ propagates this through failures, and $ν ≪ π ≪ ρ$ handles successes. Both success and failure endpoint-pair measures are absolutely continuous with respect to $ρ (d x) Q (x, d y)$ , because their endpoint kernels are bounded by $Q$ . Thus this bridge identity holds on every transition actually sampled.

5.1F5F6F9step 4.1step 4.2step 1.2construct

Let $U_{j}$ consist of the $j$ th iid core and an independent infinite sequence of bridge uniforms, using [F5]. For a nonterminal core transition both skeleton endpoints lie in $U_{j}$ . For its terminal successful transition the starting endpoint lies in $U_{j}$ and the ending endpoint is the first state in $U_{j + 1}$ ; use a uniform in $U_{j}$ for its bridge. Concatenate each transition's starting state and its $m - 1$ intermediate states, excluding its endpoint. This gives a measurable finite word $W (U_{j}, U_{j + 1})$ of length $L (U_{j}) = m ℓ (U_{j})$ . Concatenate also the finite initial segment from step 4.1, with its terminal bridge ending at the first state's value in $U_{0}$ . Success and failure endpoint kernels add to $Q$ , and step 4.2's bridge kernel then restores the original $m$ -step path law. Iterating these identities gives the original full $P$ -chain law from $μ$ . Thus these words are the claimed delayed calendar tours in a chain copy, with rewards and absolute envelopes measurable functions of two adjacent seeds.

5.2F6F8step 2.1step 3.1step 4.1algebra

The invariant identity is $π = π K_{0} + p ν$ . Iteration gives $π = p \sum_{j = 0}^{n - 1} ν K_{0}^{j} + π K_{0}^{n}$ . The remainder's total mass is the stationary probability of no successful coin in its first $n$ opportunities, which tends to zero by step 3.1 and monotone convergence. Therefore the potential measure $η = \sum_{j \geq 0} ν K_{0}^{j}$ equals $π / p$ . It counts every core state before its fresh coin is sampled, including the state of the terminal successful coin. In particular $E ℓ = η (S) = 1 / p$ and $E L = m / p$ .

6.1F6F7F8step 4.2step 5.1step 5.2

For nonnegative measurable $b$ , set $c_{b} (x) = \sum_{i = 0}^{m - 1} P^{i} b (x)$ . The event that a skeleton state is counted in the core is determined before its fresh coin and bridge are sampled. Averaging that coin, its endpoint and bridge makes the conditional expected calendar-block cost $c_{b} (x)$ , because their endpoint mixture is $Q$ and their restored block law is the original one. Hence [F6] and step 5.2 give $E \sum_{t in one full tour} b (X_{t}) = η (c_{b}) = \frac{π (c_{b})}{p} = \frac{m}{p} π (b) = μ_{L} π (b) .$ This includes the cost inside the terminal successful bridge. Apply the identity to the positive and negative parts of $f$ ; $π ∣ f ∣ < \infty$ follows from its stated $v$ th moment and [F7]. Thus $E Z_{0} = 0$ .

6.2F6F7F8step 2.1step 4.1step 5.1

By step 4.1, $E \exp (a L) < \infty$ for $a = (\log b) / m > 0$ . Starting just after a success, the reconstructed infinite chain has initial law $ν$ and its original $P$ -chain marginal, by step 5.1. Since $π \geq p ν$ , invariance gives $E_{ν} ∣ f (X_{t}) ∣^{v} \leq π (∣ f ∣^{v}) / p$ at every calendar time. For $1 \leq q < v$ , Minkowski and Holder [F7], first on finite sums, imply ${∥ \sum_{t = 0}^{L - 1} ∣ f (X_{t}) ∣ ∥}_{q} \leq {(\frac{π (∣ f ∣^{v})}{p})}^{1 / v} \sum_{t \geq 0} \Pr_{ν} (L > t)^{1 / q - 1 / v} < \infty .$ The exponential length moment makes the series finite; [F6] passes through the increasing finite sums. This proves every claimed spare moment, while $∣ Z_{0} ∣ \leq H_{0}$ .

7.1A1F3F4F5step 3.1step 1.2step 5.1step 6.1step 6.2

The constructed state path, its finite delay, and its iid seed sequence have a joint law on standard-Borel spaces by the explicit countable coding in step 1.2. Disintegrate that law conditional on the full state path by [F3], and evaluate the auxiliary kernel at the actual given chain path. Extend its probability space by [F5]. The original entire process is preserved, and its joint law with the seeds and delay is the constructed one. Thus the word equalities, iid seed law, occupation identity and moment conclusions hold on this extension. The initial segment is almost surely finite by step 3.1; since $f$ is finite, its absolute reward is finite, even without an initial-law moment bound.

8.1F5step 4.2step 5.1step 5.2step 6.1step 6.2step 7.1∎

Each length depends only on $U_{j}$ , so the lengths are iid with positive finite mean. Each reward and envelope depends on $(U_{j}, U_{j + 1})$ , so their joint sequence with the centered lengths is adjacent-iid, with $E Z_{0} = 0$ and all the stated moments. Disjoint seed pairs are independent, giving one-dependence without claiming iid full tours. The same $m, C, ν$ and core law were selected from $P, π, M$ , and the single bridge version in step 4.2 uses a $ρ$ dominating both the specified initial law and $π$ . Thus the same seed law and word functions work from either initial law; only the initial segment and its lifting kernel change. All properties in the Statement follow. The chosen split came from step 1.1 and never used positivity of any supplied one-step minorization, which can therefore be vacuous.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Heterogeneous bounded atom vector rows admit a logarithmic cubed Gaussian coupling

Statement

Assume AC. Fix $K \geq 1$ and $A > 0$ . There is a finite $C_{A, K}$ such that the following holds uniformly over $1 \leq d \leq 2$ , integers $1 \leq m \leq N$ , and $B \geq 1$ . Let $X_{1}, \dots, X_{m}$ be independent centered vectors with representations $X_{i} = Z_{i} + \sum_{r = 1}^{r_{i}} Y_{i r}, ∣ Y_{i r} ∣ \leq B almost surely,$ where every $r_{i}$ is finite, all displayed atom and Gaussian terms are independent, every $Y_{i r}$ is centered, and every $Z_{i}$ is a centered Gaussian vector, possibly degenerate. Suppose $I_{d} \leq D_{i} : = cov (X_{i}) \leq K I_{d} .$ There is a coupling with independent $G_{i} \sim N_{d} (0, D_{i})$ satisfying $\Pr ⁣ (\max_{1 \leq k \leq m} ∣ \sum_{i = 1}^{k} (X_{i} - G_{i}) ∣ > C_{A, K} B \log^{3} (e N)) \leq N^{- A} .$ The constants do not depend on the atom counts or individual laws. Empty atom sums and zero Gaussian terms are allowed.

The coupling laws can be chosen as a measurable probability kernel in a supplied standard Borel parameter $θ$ : if the atom laws and Gaussian covariance matrices are measurable kernels and functions of $θ$ , with the hypotheses holding for each $θ$ , there is a coupling kernel $Q_{θ} (d x, d g)$ with the stated product marginals and the displayed bound for each $θ$ . Thus a random environment with these conditional independent atom laws can first be sampled, and this coupling kernel then applied conditionally. This assertion concerns the conditional laws supplied in the hypotheses and does not assume a selection of regular conditional laws separately at each parameter.

Facts & Assumptions

Given: The atom and Gaussian representations, covariance bounds, parameters $A, K, d, m, N, B$ , and, for the kernel assertion, the supplied measurable atom laws and Gaussian covariance matrices. Matrix inequalities are quadratic form inequalities and all vector norms are Euclidean.

[A1]

AC is assumed. The Axiom of Choice

[F1]

Independent sums of centered bounded vectors and a Gaussian, of total covariance $s I_{j}$ and Gaussian covariance at least $γ s I_{j}$ , have the uniform cumulant and tilted Fourier bounds of Gaussian anchored bounded sums have uniform cumulant and Fourier bounds.

[F2]

The Gaussian anchored conditional quantile supplier constructs measurable root transports and, for joint dimension $2 d \leq 4$ , measurable transports of a fresh independent Gaussian residual conditional on the target parent. With covariance $s I$ , atom bound $b$ , and Gaussian covariance fraction $γ > 0$ , the target value error is bounded by $C_{γ} b (1 + (∣ p ∣^{2} + ∣ v ∣^{2}) / s)$ when $s \geq C_{γ} b^{2}$ and $∣ p ∣, ∣ v ∣ \leq c_{γ} s / b$ . Conditional quantile couplings for Gaussian anchored sums

[F3]

Real symmetric matrices have an orthonormal eigenbasis. Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

[F4]

A Gaussian has its covariance characteristic function; zero cross covariance implies independence for Gaussian subvectors by factorization. A positive definite Gaussian has its Gaussian density. Characteristic function of a multivariate normal law, A positive-definite multivariate normal law has its Gaussian density

[F5]

Integrating a measurable function against a supplied probability kernel is measurable, and finite products of probability kernels are probability kernels. Measurability of integration against a kernel, Finite products of probability kernels are probability kernels

[F6]

Nonnegative integrals obey Tonelli and monotone convergence. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Monotone convergence for the integral

[F7]

A kernel extension depending only on one part of an independent pair preserves independence from its complement. Conditional independence under kernel extension and conditioning

[F8]

Exponential Markov bounds, the finite union bound, and Euclidean triangle inequality apply. Markov's inequality for random variables, The finite union bound, Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

[F9]

Expectations of finite products of nonnegative functions of independent variables factor. Expectations factor over finite products of independent random variables

Proof

1.1givenA1F3F4F6F8F9algebra

For a coordinate $Y$ of one centered atom, $E ∣ Y ∣^{r} \leq B^{r - 2} E Y^{2}$ for every integer $r \geq 2$ . Absolute exponential-series integration is legitimate because $\sum_{r} ∣ h Y ∣^{r} / r! \leq e^{B ∣ h ∣}$ ; [F6] permits summing its nonnegative absolute terms, and its positive and negative parts then give the signed series identity. For $B ∣ h ∣ < 1$ , centering cancels the linear term and $r! \geq 2$ for $r \geq 2$ , so $\log E e^{h Y} \leq E e^{h Y} - 1 \leq \frac{h^{2} E Y^{2}}{2 (1 - B ∣ h ∣)} .$ A centered Gaussian coordinate has logarithmic mgf $h^{2} var (Z) / 2$ , by completing the square in its positive-variance eigencoordinates using [F3] and [F4]; zero-variance coordinates vanish. Independence multiplies the finitely many atom and Gaussian mgfs. Since their variances add to each coordinate variance of $X_{i}$ , at most $K$ , any sum of $s$ entries, also with arbitrary matching Gaussian replacements, has coordinate log mgf at most $K s h^{2} / [2 (1 - B ∣ h ∣)]$ . For a coordinate threshold $u > 0$ use $h = u / (K s + B u)$ and exponential Markov for both signs. This gives tail at most $2 \exp {- u^{2} / [2 (K s + B u)]}$ . The coordinate union bound therefore gives, uniformly in atom counts, $\Pr ⁣ (∣ \sum_{i \in I} X_{i} ∣ > C_{K} [\sqrt{s t} + B t]) \leq C_{K} e^{- c_{K} t}, t \geq 1.$ This holds for every contiguous interval in any compressed ordering.

1.2givenF3algebra

Consider a dyadic mixed row whose odd entries are independent matching Gaussians and whose even entries have their raw laws. At a node of $s \geq 4$ entries let its two child covariance matrices be $U, V$ , put $W = U + V$ , and, for its target child sums $S_{L}, S_{R}$ , define $P = S_{L} + S_{R}, H = U W^{- 1}, D = S_{L} - H P, R = U - U W^{- 1} U .$ Independence gives $cov (P) = W$ , $cov (P, D) = U - W W^{- 1} U = 0$ , and $cov (D) = R$ . The symmetric matrix $R$ equals $U W^{- 1} V$ and has inverse $V^{- 1} W U^{- 1} = U^{- 1} + V^{- 1}$ ; hence it is positive definite without assuming commutation. Since each child has $s / 2$ entries, [F3] gives $\frac{s}{2} I \leq U, V \leq \frac{K s}{2} I, s I \leq W \leq K s I, \frac{s}{4} I \leq R \leq \frac{K s}{4} I .$ The final inequalities follow by inverting, adding, and reinverting the quadratic form bounds.

2.1F1F2F3step 1.2algebra

The transformed pair $Q = \sqrt{s} diag (W^{- 1 / 2}, R^{- 1 / 2}) (P, D)$ has covariance $s I_{2 d}$ . A raw atom $x$ in its left or right child contributes respectively $\sqrt{s} (W^{- 1 / 2} x, R^{- 1 / 2} V W^{- 1} x), \sqrt{s} (W^{- 1 / 2} x, - R^{- 1 / 2} U W^{- 1} x) .$ Their norms are at most $C_{K} B$ , since $\sqrt{s} ∥ W^{- 1 / 2} ∥ \leq 1$ , $\sqrt{s} ∥ R^{- 1 / 2} ∥ \leq 2$ , and $∥ U W^{- 1} ∥, ∥ V W^{- 1} ∥ \leq K / 2$ . Each child contains half Gaussian entries, so its Gaussian covariance is at least $(s / 4) I$ , at least $γ$ times its total covariance with $γ = 1 / (2 K)$ . Congruence of quadratic forms therefore makes the Gaussian covariance of $Q$ at least $γ s I_{2 d}$ . All its bounded contributions are the independently transformed original atoms, and all Gaussian contributions combine into an independent Gaussian. Thus [F1] and [F2] apply directly to $Q$ , including its coordinate prefixes. The conditioning is on its first $d$ coordinates. Undoing the separate parent and residual whitening, whose inverse norms are bounded by $C_{K}$ , gives a transport of a fresh $E_{I} \sim N_{d} (0, R)$ to $D$ conditional on $P = p$ , with error $∣ T_{I, p} (E_{I}) - E_{I} ∣ \leq C_{K} B (1 + \frac{∣ p ∣^{2} + ∣ T_{I, p} (E_{I}) ∣^{2}}{s})$ when $s \geq C_{K} B^{2}$ and the two target norms are at most $c_{K} s / B$ . The root has the analogous transport from $N_{d} (0, W)$ with error $C_{K} B (1 + ∣ P ∣^{2} / s)$ . There is no additional factor from the atom counts or from $\sqrt{s}$ .

3.1F2F4step 1.2step 2.1constructchoosealgebra

Temporarily take a dyadic current length $q \leq 2 N$ , set $L = a \log (e N)$ , and let $q_{0}$ be the least dyadic integer at least $H_{K} B^{2} L$ , with $H_{K}$ to be fixed. Suppose $q \geq 2 q_{0}$ . Start with an independent Gaussian row having the individual covariances $D_{i}$ . Its root sum and residuals $E_{I} = V_{L} - U W^{- 1} V_{I}$ at the active nodes are mutually independent Gaussian vectors of covariances $W_{r o o t}$ and $R_{I}$ : the cross covariances vanish, and independence follows recursively from [F4]. Apply the root transport in step 2.1 and then, descending the tree, set $D_{I} = T_{I, S_{I}} (E_{I})$ and $S_{L} = U W^{- 1} S_{I} + D_{I}, S_{R} = V W^{- 1} S_{I} - D_{I}, V_{L} = U W^{- 1} V_{I} + E_{I}, V_{R} = V W^{- 1} V_{I} - E_{I} .$ Stop at length $q_{0}$ . Each fresh residual is independent of the already exposed parents, so every target split has exactly its prescribed conditional law. For Gaussians the child covariance is $U W^{- 1} U + R = U$ and the cross covariance is $U W^{- 1} V - R = 0$ , with the analogous right covariance $V$ .

4.1F4F5F6step 3.1constructalgebra

There is an explicit terminal target kernel, also measurable in any supplied environment. In a terminal mixed block let $μ$ be the product law of its raw positions, let $s (r)$ be their sum, and let $G > 0$ be the sum of the matching Gaussian-position covariances. Given the target block sum $u$ , sample the raw-position vector from $\frac{q_{G} (u - s (r)) μ (d r)}{\int q_{G} (u - s (v)) μ (d v)} .$ The denominator is finite and strictly positive everywhere by [F4]. Fill the Gaussian positions conditional on their remaining sum $u - s (r)$ : for their covariance matrices $C_{j}$ and fresh independent $h_{j} \sim N (0, C_{j})$ , put $H_{0} = \sum_{j} h_{j}$ and $g_{j} = C_{j} G^{- 1} (u - s (r)) + h_{j} - C_{j} G^{- 1} H_{0}$ . The residual vector has sum zero, zero cross covariance with $H_{0}$ , and covariance blocks $δ_{i j} C_{i} - C_{i} G^{- 1} C_{j}$ ; thus [F4] identifies its Gaussian conditional law. Multiplying the posterior by the total-sum density and integrating in $u$ proves the original raw/Gaussian product law, first on rectangles and then by monotone approximation. This verifies the terminal conditional law at every sum, without choosing regular conditional versions. Independent terminal kernels and the split laws of step 3.1 consequently recover exactly the entire mixed row product law by induction through the disjoint binary blocks. The existing Gaussian row is unchanged, and already has its required product law.

5.1F8step 1.1step 1.2step 2.1step 4.1choosealgebra

Use step 1.1 and a union bound for every dyadic target sum and every within-terminal prefix in the completed target and Gaussian rows. There are at most $C q$ such vectors. Except on probability $C_{K} q e^{- c_{K} L}$ , each target sum of size $s$ has norm at most $C_{K} (\sqrt{s L} + B L)$ . The target residual is $D_{I} = V W^{- 1} S_{L} - U W^{- 1} S_{R}$ , so its norm satisfies the same bound with a different $C_{K}$ . At active sizes $s \geq 2 q_{0}$ , choose $H_{K}$ large enough that these target norms are at most $c_{K} s / B$ ; their ratios to $s / B$ are bounded by $C_{K} / \sqrt{H_{K}} + C_{K} / H_{K}$ . Their squared norms divided by $s$ are at most $C_{K} L$ . Step 2.1 therefore bounds the root discrepancy and every residual discrepancy $δ_{I} = D_{I} - E_{I}$ by $C_{K} B L$ . The within-terminal prefix norms are at most $C_{K} (\sqrt{q_{0} L} + B L) \leq C_{K} B L$ , since $q_{0} < 2 H_{K} B^{2} L$ . No conditional concentration inside a terminal block has been assumed.

6.1F8step 1.2step 3.1step 5.1algebra

Write $e_{I} = S_{I} - V_{I}$ . Its child equations are $e_{L} = U W^{- 1} e_{I} + δ_{I}$ and $e_{R} = V W^{- 1} e_{I} - δ_{I}$ . For a prefix $P$ ending at a terminal boundary of $I$ , let $C_{P}$ be the sum of its individual covariance matrices. Its discrepancy has the exact expansion $\sum_{i \in P} (X_{i} - G_{i}) = C_{P} W_{I}^{- 1} e_{I} + \sum_{J on the boundary path} M_{J, P} δ_{J} .$ If the boundary lies in the left child, $M_{J, P} = C_{p a r t i a l l e f t} W_{J_{L}}^{- 1}$ ; if it lies in the right child, $M_{J, P} = I - C_{p a r t i a l r i g h t} W_{J_{R}}^{- 1}$ . To prove the formula, in the left case substitute the child's leading term $C_{P} U^{- 1} e_{L}$ and cancel $U^{- 1} U$ ; in the right case add $e_{L}$ to $C_{Q} V^{- 1} e_{R}$ , obtaining leading coefficient $(U + C_{Q}) W^{- 1}$ and residual coefficient $I - C_{Q} V^{- 1}$ . Induction from terminal blocks proves the expansion. This cancellation uses the order $W_{d e s c} W_{p a r e n t}^{- 1} W_{p a r e n t} W_{g r a n d p a r e n t}^{- 1}$ and never commutes two matrices. If a partial child has $r$ of its $h$ entries, its coefficient norm is at most $K r / h \leq K$ . Hence $∥ M_{J, P} ∥ \leq 1 + K$ and $∥ C_{P} W_{I}^{- 1} ∥ \leq K$ . There are at most $\log_{2} q$ boundary ancestors. The expansion and step 5.1 give terminal-boundary error $C_{K} B L \log (e q)$ ; adding its terminal prefix residual gives the same bound for all prefixes.

7.1F8step 1.1step 6.1algebra

If $q < 2 q_{0}$ , couple a raw or mixed row independently to a matching Gaussian row. Their unconditional concentration bounds from step 1.1 and a union bound over prefixes give maximum discrepancy at most $C_{K} (\sqrt{q L} + B L) \leq C_{K} B L$ except on probability $C_{K} q e^{- c_{K} L}$ . This covers the small sizes, including a row of one entry, and large $B$ . Thus in all cases a mixed row can be coupled directly to an already realized matching Gaussian row with stage error at most $C_{K} B L \log (e q)$ and bad probability at most $C_{K} q e^{- c_{K} L}$ .

8.1F5F7step 3.1step 4.1step 5.1step 6.1step 7.1construct

Pad the initial length $m$ to dyadic $q < 2 m$ by independent $N_{d} (0, I_{d})$ entries; for $m = 1$ take $q = 1$ . These added laws have empty atom sums and preserve all covariance bounds. Let $R^{(0)}$ be the matching independent Gaussian row, saved as the final reference row, and initially mark every position active. Inductively, the current full row consists of independent frozen raw entries and an independent active Gaussian row, all with the designated covariances. When its active length is at least $2 q_{0}$ , use that active Gaussian row and fresh terminal randomness in steps 3.1--6.1 to construct the target mixed row, with designated raw laws in even slots and matching Gaussian laws in odd slots. Replace the entire active row by this target row, including its odd coordinates, to obtain $R^{(j + 1)}$ from $R^{(j)}$ . Freeze the target evens and use the target Gaussian odds as the next active row. The extension kernel depends only on the previous active row, so [F7] makes the entire target independent of the previously frozen complement; its product marginal from step 4.1 makes its new raw evens independent of its Gaussian odds. This proves the induction. At active length below $2 q_{0}$ , replace every remaining active coordinate by independent designated raw entries using step 7.1 and stop. After at most $C \log (e N)$ stages, the final row $R^{(J)}$ has the intended raw product law. The saved row $R^{(0)}$ retains its Gaussian product law, although subsequent active Gaussian rows need not equal its corresponding coordinates.

9.1F8step 6.1step 7.1step 8.1choosealgebra

Write $Δ_{j} (k) = \sum_{i \leq k} (R_{i}^{(j + 1)} - R_{i}^{(j)})$ . Frozen entries do not change at stage $j$ , and every original prefix meets its active positions in an initial compressed prefix. Thus the stage estimate in steps 6.1--7.1 bounds $\max_{k} ∣ Δ_{j} (k) ∣$ by $C_{K} B L \log (e N)$ outside its stage bad event; this includes all changes to target Gaussian odds. The exact identity $\sum_{i \leq k} (R_{i}^{(J)} - R_{i}^{(0)}) = \sum_{j = 0}^{J - 1} Δ_{j} (k)$ and triangle inequality therefore give total error at most $C_{A, K} B \log^{3} (e N)$ . Each stage has the Gaussian input marginal and mixed target marginal used in its estimate by step 8.1, so its unconditional bad probability remains at most $C_{K} q_{j} e^{- c_{K} L}$ , where $q_{j} \leq 2 N$ is its active length. With the same ambient $L = a \log (e N)$ , a union bound over at most $C \log (e N)$ stages gives $C_{K} N \log (e N) e^{- c_{K} a \log (e N)}$ . Choose $a$ sufficiently large depending only on $A, K$ to make this at most $N^{- A}$ for all $N \geq 1$ . Taking $X = R^{(J)}$ , $G = R^{(0)}$ and restricting to the first $m$ entries proves the numerical assertion.

10.1A1F2F3F4F5F6step 1.2step 2.1step 4.1step 9.1constructalgebra∎

For the parameter assertion, every entry law is a measurable kernel obtained by applying addition to the finite product of its supplied atom and Gaussian kernels, using [F5]. Covariance entries are measurable integrals. In dimensions one and two positive definite inverse square roots are continuous functions of their matrices: in dimension two, for $C > 0$ , $C^{1 / 2} = (C + \sqrt{\det C} I) / \sqrt{tr C + 2 \sqrt{\det C}}$ , as verified in an eigenbasis by [F3]; the semidefinite zero case is defined by zero and the formula extends continuously. Thus the Gaussian starts and all transforms in steps 1.2, 2.1 depend measurably on the parameter. Mixed transformed joint and prefix densities are Gaussian convolution integrals of their raw product kernels, so are jointly measurable by [F5] and strictly positive. Their conditional CDFs are measurable density-ratio integrals. The coordinate quantiles in [F2] are measurable rational infima of those CDFs; hence its root and residual maps are jointly measurable in the parameter and all spatial inputs. The terminal posterior in step 4.1 is likewise a measurable kernel with positive measurable denominator, followed by the explicit Gaussian bridge kernel. Finite kernel products and successive extensions preserve measurability: for one extension define its law by integrating the next kernel against the preceding law; [F5] gives measurable evaluations and [F6] gives countable additivity by monotone convergence of disjoint indicator sums. Iterating this finite construction gives a coupling kernel $Q_{θ}$ , and every previous marginal and probability bound holds at each parameter. No pointwise selection of conditional-law versions or coupling laws is used.

Source notes

The source supplies the dyadic conditional-quantile architecture. The heterogeneous covariance formulas in steps 1.2 and 2.1, the stopped Gaussian-anchor replacement, and the parameterwise terminal posterior are proved locally. The bound is deliberately weaker than the source's optimized strong approximation estimates. Whole grouped entries need not be bounded: the boundedness hypothesis concerns their independent centered atoms.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Sufficient MCMC CLT regimes

Statement

Assume AC. Let $(X, X)$ be countably generated, let $P$ be a $ϕ$ -irreducible probability kernel (Phi-irreducibility for a general-state Markov kernel, Measure kernel and probability kernel), aperiodic (Aperiodicity for a general-state Markov kernel), and Harris recurrent relative to $ϕ$ (Harris recurrence relative to an irreducibility measure) with invariant probability $π$ , and let $(X_{k})_{k \geq 0}$ be a stationary $P$ -chain whose canonical left shift preserves its path law and whose shift-invariant measurable events have probability zero or one. We take the initial law to be $π$ , which implies stationarity by Invariant initial law makes a Markov chain stationary. Let $f : X \to R$ be Borel with $π (f) = 0$ . Define $σ_{f}^{2} = π (f^{2}) + 2 \sum_{k \geq 1} {Cov}_{π} (f (X_{0}), f (X_{k}))$ only when the covariance series is absolutely convergent and the CLT is established, as in Long-run variance for a stationary Markov chain.

Each of the following six conditions is sufficient for absolute convergence of the covariance series and $\frac{1}{\sqrt{n}} \sum_{k = 0}^{n - 1} f (X_{k}) \Rightarrow N (0, σ_{f}^{2}) :$

$P$ has polynomial total-variation rate of order $m > 1$ with an integrable envelope (Polynomial ergodicity with an integrable rate envelope), and $f$ is essentially bounded under $π$ (there is $B < \infty$ with $∣ f ∣ \leq B$ $π$ -almost everywhere).
$P$ has polynomial total-variation rate of order $m > 0$ with an integrable envelope (Polynomial ergodicity with an integrable rate envelope), $f \in L^{2 + δ} (π)$ for some $δ > 0$ (The space $L^{p} (μ)$ as the quotient by null functions), and $m δ > 2 + δ$ .
$P$ is geometrically ergodic (Geometric and uniform ergodicity for MCMC CLTs) and $f \in L^{2 + δ} (π)$ for some $δ > 0$ (The space $L^{p} (μ)$ as the quotient by null functions).
$P$ is geometrically ergodic (Geometric and uniform ergodicity for MCMC CLTs) and $π (f^{2} \log^{+} ∣ f ∣) < \infty$ , with $0 \log^{+} 0 : = 0$ .
$P$ is geometrically ergodic (Geometric and uniform ergodicity for MCMC CLTs), reversible with respect to $π$ in the symmetric-flow sense (General-state reversibility by symmetric flow measure), and $f \in L^{2} (π)$ (The space $L^{p} (μ)$ as the quotient by null functions).
$P$ is uniformly ergodic (Geometric and uniform ergodicity for MCMC CLTs) and $f \in L^{2} (π)$ (The space $L^{p} (μ)$ as the quotient by null functions).

In case 4, the displayed logarithmic moment implies $f \in L^{2} (π)$ : on ${∣ f ∣ \leq e}$ one has $f^{2} \leq e^{2}$ , and on ${∣ f ∣ > e}$ one has $f^{2} \leq f^{2} \log ∣ f ∣$ .

When $σ_{f}^{2} = 0$ , the zero-variance lemma Zero covariance long-run variance gives a degenerate CLT under absolute summability gives the degenerate normal limit from absolute covariance summability. Finite variance of $f$ alone is not one of the sufficient conditions.

Facts & Assumptions

Given: AC, the stationary Harris-ergodic chain and centered real function in the Statement, and any one of its six sufficient regimes.

[A1]

The Axiom of Choice is assumed by the local probability, conditional-expectation and sequence-CLT suppliers. The Statement already includes this hypothesis.

[F1]

A measurable observation does not increase stationary mixing coefficients makes $Y_{k} = f (X_{k})$ stationary and bounds each of its mixing coefficients by the corresponding chain coefficient.

[F2]

Polynomial ergodicity with an integrable rate envelope gives the pointwise polynomial total-variation bound with $π (M) < \infty$ . Geometric and uniform ergodicity for MCMC CLTs gives the geometric bound with finite measurable envelope, or with bounded envelope in the uniform case.

[F3]

The bounded-future Markov property and iterated-kernel identity are Markov property for bounded future path functionals and Chapman-Kolmogorov equations. The event-supremum convention is Total variation distance for probability laws, and known factors can be taken out using Taking out what is known.

[F4]

Strong-mixing CLTs under bounded, moment, and logarithmic-moment conditions proves absolute covariance summability and the normal limit for bounded observations with summable alpha, for the stated $L^{2 + δ}$ alpha-summability condition, or for exponential alpha with the logarithmic moment.

[F5]

Harris geometric ergodicity admits an integrable total-variation envelope supplies an integrable geometric envelope under the stated Harris, irreducibility and aperiodicity hypotheses. A total-variation rate bounds strong-mixing coefficients then bounds alpha by that geometric rate.

[F6]

Geometric ergodicity and reversibility give exponential rho-mixing gives exponential rho for a stationary reversible geometrically ergodic chain under the stated recurrence hypotheses. A rho-mixing CLT for square-integrable stationary sequences proves the covariance-summable $L^{2}$ CLT when rho is summable.

[F7]

Uniform ergodicity bounds the uniform-mixing coefficient gives exponential phi under uniform ergodicity. A uniform-mixing CLT for square-integrable stationary sequences proves the covariance-summable $L^{2}$ CLT when the square roots of phi are summable.

[F8]

Zero covariance long-run variance gives a degenerate CLT under absolute summability handles zero covariance long-run variance by a finite covariance expansion. Long-run variance for a stationary Markov chain identifies the covariance sum only after absolute convergence and the CLT have been established.

[F9]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges proves convergence of $\sum_{j \geq 0} q^{j}$ for every real $0 \leq q < 1$ .

Proof

Given: The hypotheses and any one of the six regimes in the Statement.

1.1A1F1givenalgebra

Put $Y_{k} = f (X_{k})$ . This is a centered stationary real sequence by [F1]. It is square-integrable in every case: boundedness suffices in case 1, the power moment suffices in cases 2 and 3 on the probability space, and cases 5 and 6 assume $L^{2}$ . In case 4, split at $∣ f ∣ = e$ : on the bounded part $f^{2} \leq e^{2}$ , and on the other part $f^{2} \leq f^{2} \log ∣ f ∣$ , so the stated logarithmic moment implies $L^{2}$ . All sequence suppliers below therefore have their centering, stationarity and second-moment premises.

1.2A1F1F2F3algebra

Suppose a pointwise rate $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) a_{n}$ is available with $π (M) < \infty$ . For any past event $A \in σ (X_{0}, \dots, X_{j})$ and separated future event $B \in σ (X_{j + n}, X_{j + n + 1}, \dots)$ , [F3] supplies $0 \leq h \leq 1$ with $E [1_{B} ∣ σ (X_{0}, \dots, X_{j})] = P^{n} h (X_{j})$ and $P (B) = π (h)$ . Approximating $h$ uniformly by dyadic averages of its superlevel indicators gives $∣ P^{n} h (x) - π (h) ∣ \leq ∥ P^{n} (x, \cdot) - π ∥_{T V}$ : each indicator discrepancy is bounded by the defining event supremum. Consequently $∣ P (A \cap B) - P (A) P (B) ∣ = ∣ E [1_{A} (P^{n} h (X_{j}) - π (h))] ∣ \leq π (M) a_{n} .$ Taking the suprema and then applying [F1] gives $α_{Y} (n) \leq α_{X} (n) \leq π (M) a_{n}$ . No measurability of an uncountable total-variation supremum is needed, because its measurable envelope bounds each event discrepancy pointwise.

2.1F2F4F9step 1.1step 1.2givenalgebra

In case 1, take $a_{n} = n^{- m}$ in step 1.2. Since $m > 1$ , grouping $2^{j} \leq n < 2^{j + 1}$ bounds each block of $\sum_{n} n^{- m}$ by $2^{- j (m - 1)}$ ; [F9] makes the resulting geometric series finite. Thus $\sum_{n} α_{Y} (n) < \infty$ . The bounded branch of [F4] gives absolute covariance summability and the asserted normal limit for the normalized sums of $Y$ . This uses the whole approved range $m > 1$ .

2.2F2F4F9step 1.1step 1.2givenalgebra

In case 2, put $θ = δ / (2 + δ)$ . Step 1.2 gives $α_{Y} (n)^{θ} \leq π (M)^{θ} n^{- m θ}$ . The exact assumption $m δ > 2 + δ$ says $m θ > 1$ , so the same dyadic argument and [F9] make these bounds summable. The power-moment branch of [F4] proves the same covariance and distribution conclusions.

2.3F1F4F5F9step 1.1givenalgebra

In cases 3 and 4, [F5] replaces the possibly nonintegrable original envelope by a finite integrable geometric envelope with some rate $a \in (0, 1)$ . Thus $α_{Y} (n) \leq C a^{n}$ by [F1] and [F5]. In case 3, every positive power of this bound is summable, so the $L^{2 + δ}$ branch of [F4] applies. In case 4, its exponential/logarithmic-moment branch applies directly. Each yields absolute covariance summability and the normal limit without strengthening the given moment assumption.

2.4A1F1F6F9step 1.1givenalgebra

In case 5, reversibility and the geometric-rate assumptions allow [F6] to give $ρ_{X} (n) \leq C r^{n}$ for some $r < 1$ . By [F1], $ρ_{Y} (n) \leq ρ_{X} (n)$ , so its series is summable. The $L^{2}$ rho-mixing theorem in [F6] proves the required covariance-summable CLT.

2.5A1F1F7F9step 1.1givenalgebra

In case 6, [F7] gives $ϕ_{X} (n) \leq C r^{n}$ for some $r < 1$ . By [F1], $\sum_{n} \sqrt{ϕ_{Y} (n)} \leq \sqrt{C} \sum_{n} r^{n / 2} < \infty$ . The uniform-mixing $L^{2}$ theorem in [F7] therefore gives the same conclusions.

3.1F1F4F6F7F8step 2.1step 2.2step 2.3step 2.4step 2.5algebra∎

In all six cases the covariance series is absolutely convergent, its sum is finite and nonnegative, and the normalized sum converges to the centered normal with precisely that covariance sum. If the sum is zero, [F8] also proves the degenerate limit directly in $L^{2}$ , so no division by an asymptotic variance is made. Stationarity makes $\sum_{k = 0}^{n - 1} Y_{k}$ and $\sum_{k = 1}^{n} Y_{k}$ equal in law. The established limit is therefore exactly the displayed limit in the Statement, and only now does [F8] identify its variance as $σ_{f}^{2}$ .

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Zero-variance geometric Markov sums have a uniform strong rate

Statement

Assume AC. Let $P$ be a probability kernel on an arbitrary measurable space $(X, X)$ with invariant probability $π$ . Suppose its stationary Markov chain has strong-mixing coefficients $α_{X} (n) \leq C r^{n}$ for $n \geq 1$ , with $0 \leq C < \infty$ and $0 < r < 1$ . Let $f : X \to R$ be finite measurable with $π f = 0$ , $π ∣ f ∣^{q} < \infty$ for a finite $q > 2$ , and suppose the absolutely convergent covariance sum is zero: $v : = π (f^{2}) + 2 \sum_{n \geq 1} {Cov}_{π} (f (X_{0}), f (X_{n})) = 0.$ Then for every $2 < p < q$ , under the stationary law, $\max_{0 \leq j \leq n} ∣ \sum_{k = 0}^{j - 1} f (X_{k}) ∣ = o (n^{1 / p}) almost surely .$ The empty sum at $j = 0$ is zero. No ergodicity assumption is needed beyond the stated stationary mixing bound.

If additionally $∥ P^{n} (x, \cdot) - π ∥_{T V} \to 0$ for every $x \in X$ , the same assertion holds under every initial probability law, without a moment assumption on that law. In particular, it holds under every initial law whenever $P$ has an everywhere finite measurable common geometric total-variation envelope in the sense of Geometric and uniform ergodicity for MCMC CLTs and $f$ has the stated moment, centering and zero covariance sum. No positive one-step minorization, irreducibility or Harris hypothesis is added to this last sufficient condition.

Facts & Assumptions

Given: AC, the stationary geometric strong-mixing hypotheses, finite centered real $f \in L^{q} (π)$ , the zero covariance sum, and any fixed $2 < p < q$ . For the all-start conclusion assume also the stated pointwise total-variation convergence; for the final sufficient condition assume the everywhere geometric envelope instead.

[A1]

AC is assumed. The Axiom of Choice

[F1]

Under the stationary geometric strong-mixing and moment hypotheses, there is a finite measurable $g \in L^{p} (π)$ with $f = g - P g$ almost everywhere; the covariance series is absolutely convergent and $n^{- 1} E_{π} (\sum_{k = 0}^{n - 1} f (X_{k}))^{2} \to v$ . Geometric strong mixing gives an Lp Poisson solution

[F2]

For $g \in L^{2} (π)$ , $D_{k + 1} = g (X_{k + 1}) - P g (X_{k})$ is square-integrable with $E [D_{k + 1} ∣ F_{k}] = 0$ , where $F_{k} = σ (X_{0}, \dots, X_{k})$ . Kernel conditional expectations for square-integrable Markov observables

[F3]

On a probability space $L^{p} \subset L^{2}$ for $p > 2$ ; Cauchy–Schwarz bounds products of $L^{2}$ variables. Conditional expectation preserves means, and a known factor can be taken out when the products are integrable. Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$ , Cauchy-Schwarz for random variables, Basic algebra and order properties of conditional expectation, Taking out what is known

[F4]

Monotone convergence passes increasing finite nonnegative sums through expectation; summable event probabilities imply only finitely many occurrences almost surely. Monotone convergence for the integral, First Borel-Cantelli lemma for events

[F5]

Under AC the canonical Markov path law exists for every initial law. For a bounded product-measurable path functional $H$ , $h (x) = E_{x} H (X_{0}, X_{1}, \dots)$ is measurable and $E [H (X_{n}, X_{n + 1}, \dots) ∣ F_{n}] = h (X_{n})$ . An invariant initial law makes the chain stationary. Canonical Markov chain on path space, Markov property for bounded future path functionals, Invariant initial law makes a Markov chain stationary

[F6]

Total variation is $∥ μ - ν ∥_{T V} = \sup_{A} ∣ μ (A) - ν (A) ∣$ . Total variation distance for probability laws

[F7]

An invariant kernel with an everywhere finite measurable common geometric total-variation envelope admits such an envelope integrable under $π$ , possibly at a slower geometric rate. That integrable bound implies $α_{X} (n) \leq π (M) r^{n}$ for the stationary chain. Harris geometric ergodicity admits an integrable total-variation envelope, A total-variation rate bounds strong-mixing coefficients, Geometric and uniform ergodicity for MCMC CLTs

Proof

1.1A1F1F2F3givenalgebra

Apply [F1] to obtain $g \in L^{p} (π)$ with $f = g - P g$ almost everywhere; choose its finite measurable version as in that supplier. By [F3], $g \in L^{2}$ . Stationarity makes the Poisson identity hold at every $X_{k}$ simultaneously off a countable union of null events. By [F2], on this common event $f (X_{k}) = g (X_{k}) - g (X_{k + 1}) + D_{k + 1}$ . Summing gives $S_{n} : = \sum_{k = 0}^{n - 1} f (X_{k}) = R_{n} + M_{n}$ , with $R_{n} = g (X_{0}) - g (X_{n})$ and $M_{n} = \sum_{k = 0}^{n - 1} D_{k + 1}$ . No ergodic theorem or central limit theorem has been used.

2.1F1F2F3step 1.1givenalgebra

Put $τ^{2} = E_{π} D_{1}^{2}$ . Stationarity of transition pairs gives $E D_{j}^{2} = τ^{2}$ for every $j \geq 1$ . If $i < j$ , then $D_{i}$ is $F_{j - 1}$ -measurable, its product with $D_{j}$ is integrable by [F3], and [F2]–[F3] give $E [D_{i} D_{j}] = E [D_{i} E (D_{j} ∣ F_{j - 1})] = 0$ . Thus $E M_{n}^{2} = n τ^{2}$ . The inequality $(a - b)^{2} \leq 2 a^{2} + 2 b^{2}$ and stationarity give $E R_{n}^{2} \leq 4 ∥ g ∥_{2}^{2}$ . Cauchy–Schwarz [F3] and step 1.1 imply $∣ n^{- 1} E S_{n}^{2} - τ^{2} ∣ \leq 4 ∥ g ∥_{2}^{2} / n + 4 ∥ g ∥_{2} τ / \sqrt{n} \to 0$ . By [F1], the same normalized second moments tend to $v = 0$ , so $τ^{2} = 0$ . A nonnegative variable of zero expectation is zero almost surely; applying this to each $D_{j}^{2}$ and taking the countable intersection shows that all $D_{j}$ vanish simultaneously. Hence $S_{n} = g (X_{0}) - g (X_{n})$ for every $n$ on one probability-one event.

3.1F4step 1.1step 2.1algebra

Fix $ε > 0$ . Pointwise, the number of positive integers $k$ with $∣ g (x) ∣^{p} > ε^{p} k$ is at most $ε^{- p} ∣ g (x) ∣^{p}$ . Stationarity and [F4], applied first to finite sums and then to their increasing limit, therefore give $\sum_{k \geq 1} P_{π} (∣ g (X_{k}) ∣ > ε k^{1 / p}) = \sum_{k \geq 1} π (∣ g ∣^{p} > ε^{p} k) \leq ε^{- p} π ∣ g ∣^{p} < \infty$ . Borel–Cantelli [F4], intersected over positive rational $ε$ , yields $g (X_{k}) = o (k^{1 / p})$ almost surely. On that event, for every $ε > 0$ the sufficiently late terms obey $∣ g (X_{k}) ∣ \leq ε k^{1 / p} \leq ε n^{1 / p}$ for $k \leq n$ , while the maximum of the finitely many earlier terms divided by $n^{1 / p}$ tends to zero. Thus $\max_{k \leq n} ∣ g (X_{k}) ∣ = o (n^{1 / p})$ . Step 2.1 gives $\max_{0 \leq j \leq n} ∣ S_{j} ∣ \leq ∣ g (X_{0}) ∣ + \max_{j \leq n} ∣ g (X_{j}) ∣ = o (n^{1 / p})$ , proving the stationary conclusion.

4.1givenconstructalgebrastep 3.1

For the all-start part, define the original path event $E = {(x_{k})_{k \geq 0} : n^{- 1 / p} \sum_{k = 0}^{n - 1} f (x_{k}) \to 0}$ . It is product-measurable, since the condition is a countable combination of inequalities for finite measurable sums. It is invariant under deleting the first coordinate: if $s_{n} = \sum_{k = 0}^{n - 1} f (x_{k})$ , the shifted sum is $s_{n + 1} - f (x_{0})$ , and the reverse identity is $s_{n} = f (x_{0}) + (s_{n} - f (x_{0}))$ with the second term the shifted sum of length $n - 1$ . Since $f (x_{0})$ is finite and $((n + 1) / n)^{1 / p}$ and $((n - 1) / n)^{1 / p}$ tend to one, either normalized convergence implies the other. Iterating gives invariance under deleting any fixed finite prefix. Also, on every real path, $s_{n} = o (n^{1 / p})$ is equivalent to $\max_{j \leq n} ∣ s_{j} ∣ = o (n^{1 / p})$ : the reverse direction is immediate, and the forward direction follows by discarding a finite initial segment and bounding all later $∣ s_{j} ∣$ by $ε j^{1 / p}$ . Thus it suffices to transfer the event $E$ , rather than any Poisson identity outside a stationary full-measure set.

5.1A1F5step 3.1step 4.1

Let $h (x) = P_{x} (E)$ . By [F5], $h$ is measurable and $0 \leq h \leq 1$ . The canonical stationary chain has the same path law as the stationary chain already considered, so step 3.1 gives $π (h) = P_{π} (E) = 1$ ; this equality also follows by applying [F5] at time zero and taking expectations. For every state $x$ and integer $n \geq 1$ , prefix-deletion invariance and [F5] give $h (x) = E_{x} 1_{E} = E_{x} [h (X_{n})] = P^{n} h (x)$ . These are identities for each canonical starting state, not simultaneous versions of a conditional expectation over an uncountable family of starts.

6.1F5F6step 4.1step 5.1givenconstruct

For any probability measures $μ, ν$ and measurable $0 \leq h \leq 1$ , let $h_{m} = 2^{- m} \sum_{j = 1}^{2^{m}} 1_{{h \geq j 2^{- m}}}$ . Then $∣ h_{m} - h ∣ \leq 2^{- m}$ and [F6] gives $∣ μ (h_{m}) - ν (h_{m}) ∣ \leq ∥ μ - ν ∥_{T V}$ . Letting $m \to \infty$ proves the same bound for $h$ . Under the additional pointwise total-variation assumption, apply it to $μ = P^{n} (x, \cdot)$ and $ν = π$ . Step 5.1 gives $h (x) = P^{n} h (x) \to π (h) = 1$ for every $x$ , hence $h (x) = 1$ everywhere. For any initial law $μ$ , [F5] at time zero gives $P_{μ} (E) = \int h d μ = 1$ . The deterministic equivalence in step 4.1 transfers the uniform maximal rate as well. No moment or support condition on $μ$ is needed.

7.1A1F1F2F3F5F7step 1.1step 2.1step 3.1step 6.1given∎

Finally suppose the everywhere common geometric bound in the Statement holds. Since its envelope is finite at every state, it implies $P^{n} (x, \cdot) \to π$ in total variation for every $x$ . By [F5] the canonical invariant-initial-law chain is stationary. The stronger arbitrary-measurable-space clause of [F7] supplies a finite integrable geometric envelope, and its mixing implication supplies the stationary geometric strong-mixing bound needed by [F1]. Therefore the proof in steps 1.1, 2.1, 3.1, 4.1, 5.1, 6.1 applies to the stated $f$ and zero covariance sum under every initial law. The zero function, a one-state chain, empty partial sums and zero martingale variance are all included; an empty state space admits no invariant probability and does not satisfy the hypotheses. Strict $2 < p < q$ is used by [F1] and the $L^{2}$ comparison. AC is inherited precisely through [F1], [F2], [F3], [F5] and [F7]. This completes both the stationary result and the asserted all-start sufficient conditions.

Remarks

This is the zero-variance rate only. For a moment $π ∣ f ∣^{2 + δ + ε} < \infty$ with $δ, ε > 0$ , choose $p$ strictly between $2 + δ$ and $2 + δ + ε$ . The proved $o (n^{1 / p})$ rate is stronger than $O (n^{1 / (2 + δ)} \log n)$ . It therefore supplies the degenerate error estimate needed in a zero-Brownian-coefficient branch without using regeneration, a martingale CLT, KMT, or a nondegenerate SIP. It establishes no positive-variance Brownian coupling.

The Jones and Doukhan–Massart–Rio full PDFs were retrieved, and the source locations specified above were inspected. Their covariance and mixing methods document provenance; the bounded-dual Poisson supplier, zero-martingale variance comparison, integrated-tail rate and deletion-invariant arbitrary-start transfer are all proved locally here or in the explicit mathematical dependencies.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Finite-moment iid vectors admit a strong Gaussian approximation

Statement

Assume AC. Let $X_{1}, X_{2}, \dots$ be iid centered vectors in $R^{2}$ with $E ∥ X_{1} ∥^{q} < \infty$ for some $q > 2$ . For any fixed real numbers $2 < p < a < q$ , on an extension preserving the original process there are independent Gaussian vectors $G_{i} \sim N_{2} (0, Σ)$ , where $Σ = E X_{1} X_{1}^{T}$ , such that almost surely $\max_{1 \leq k \leq n} ∥ \sum_{i = 1}^{k} (X_{i} - G_{i}) ∥ = O (n^{1 / a} [\log (e n)]^{3}) = o (n^{1 / p}) .$ The extension can also carry a standard two-dimensional Brownian motion $B$ with $\sum_{i = 1}^{k} G_{i} = Σ^{1 / 2} B (k)$ for every integer $k \geq 0$ . Thus, in particular, the partial-sum Brownian approximation error is $O (n^{1 / p} \log n)$ almost surely. Singular covariance, including $Σ = 0$ , is permitted. Neither $B$ nor $(G_{i})$ is required to be independent of $(X_{i})$ .

Facts & Assumptions

Given: AC, the centered iid vectors $X_{i}$ , their finite $q$ th absolute moment, and fixed real exponents $2 < p < a < q$ .

[A1]

AC is assumed as in The Axiom of Choice and the construction interfaces below.

[F1]

For centered iid vectors of dimension one or two with covariance $I$ and norm at most $R \geq 1$ , the bounded-vector Gaussian coupling has a bound $C_{A} R [\log (e N)]^{3}$ on the maximum partial-sum discrepancy, with exceptional probability at most $C_{A} N^{- A}$ for every fixed $A > 0$ . Constants are uniform in $R, N$ and the bounded law. Bounded independent vector rows admit a logarithmic cubed Gaussian coupling

[F2]

A real symmetric matrix has an orthonormal eigenbasis; in particular positive definite matrices have square roots and inverse square roots obtained in that basis. Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

[F3]

Events with summable probabilities occur only finitely often almost surely. The first Borel-Cantelli lemma for measures

[F4]

Finite-dimensional Euclidean spaces are standard Borel, and a joint probability law on two standard-Borel spaces has a conditional probability kernel for one coordinate given the other. Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel, Disintegration of a joint law on standard borel spaces

[F5]

Supplied history-dependent probability kernels, with an arbitrary measurable initial probability space, admit an iterated countable extension preserving their finite-prefix laws. Ionescu-Tulcea construction of a Markov chain

[F6]

A standard normal variable satisfies $\Pr (∣ Z ∣ > t) \leq 2 e^{- t^{2} / 2}$ for $t \geq 1$ , by the Mills upper bound and symmetry. Two-sided Mills bounds for the standard normal tail

[F7]

Gaussian vectors admit their specified covariance, including singular covariance, by linear transformation of independent standard-normal coordinates; finite independent Gaussian vectors form a jointly Gaussian stack. Multivariate normal law, including singular covariance, A finite stack of independent Gaussian vectors is jointly Gaussian

[F8]

Independent Brownian copies exist in each finite dimension, and countable products of probability spaces exist under the choice principles supplied by AC. Existence and scaling of $d$ -dimensional Brownian motion, Assuming countable and dependent choice, countable products of arbitrary probability spaces

[F9]

Zero cross-covariance between jointly Gaussian blocks implies their independence. Standard vector Brownian motion is characterized by continuous paths, zero start and independent normal vector increments of the specified variances. Uncorrelated jointly Gaussian blocks are independent, $d$ -dimensional Brownian motion

Proof

1.1A1F2F8givenconstructalgebra

Put $V = ran Σ$ . For each vector $v$ in a deterministic finite orthonormal basis of $\ker Σ$ , $E (v^{T} X_{i})^{2} = 0$ , so $v^{T} X_{i} = 0$ almost surely. Intersect over that basis and the countably many indices to get one full event on which every $X_{i} \in V$ . If $V = {0}$ , take $G_{i} = 0$ and adjoin an independent standard two-dimensional Brownian motion by [F8]; both approximation errors vanish. Otherwise use a deterministic orthonormal eigenbasis of $Σ$ on $V$ and work in dimension $d = 1$ or $2$ , preserving Euclidean norms. Its covariance, still denoted $Σ$ , is positive definite, with smallest eigenvalue $λ > 0$ by [F2]. The final construction will be mapped back by the same isometry.

2.1F3step 1.1givenconstructalgebra

Let $M = E ∥ X_{1} ∥^{q}$ , $N_{m} = 2^{m}$ , $I_{m} = {2^{m}, \dots, 2^{m + 1} - 1}$ , and $L_{m} = N_{m}^{1 / a}$ . Set $μ_{m} = E [X_{1} 1_{{∥ X_{1} ∥ \leq L_{m}}}]$ and $W_{m, i} = X_{i} 1_{{∥ X_{i} ∥ \leq L_{m}}} - μ_{m}$ for $i \in I_{m}$ . The pointwise bound $x^{s} 1_{{x > L}} \leq L^{s - q} x^{q}$ for $0 \leq s \leq q$ gives $\Pr (\exists i \in I_{m} : ∥ X_{i} ∥ > L_{m}) \leq M N_{m} L_{m}^{- q} = M 2^{- m (q / a - 1)}$ . Since $q > a$ , this is summable in $m$ , and [F3] gives no discarded observations in every sufficiently large block almost surely. Also $E X_{1} = 0$ implies $∥ μ_{m} ∥ \leq M L_{m}^{1 - q}$ . On a block without discarded observations the maximum partial-block difference between $X_{i}$ and $W_{m, i}$ is at most $M N_{m} L_{m}^{1 - q} = M N_{m}^{1 + (1 - q) / a} = O (N_{m}^{1 / a})$ , since $a < q$ .

3.1F2step 1.1step 2.1algebra

The truncated covariance $Σ_{m} = E W_{m, i} W_{m, i}^{T}$ satisfies the exact identity $Σ - Σ_{m} = E [X_{1} X_{1}^{T} 1_{{∥ X_{1} ∥ > L_{m}}}] + μ_{m} μ_{m}^{T}$ . Hence $∥ Σ - Σ_{m} ∥_{o p} \leq M L_{m}^{2 - q} + M^{2} L_{m}^{2 - 2 q} \leq C L_{m}^{2 - q} \to 0$ . For all sufficiently large deterministic $m$ , its eigenvalues are therefore bounded below by $λ / 2$ and above by a fixed constant. The standardized truncated vectors $Z_{m, i} = Σ_{m}^{- 1 / 2} W_{m, i}$ are centered iid within each block, with covariance $I_{d}$ and norm bounded by $R_{m} = C_{0} L_{m} \geq 1$ , after increasing a fixed $C_{0}$ , because $∥ μ_{m} ∥ \leq E ∥ X_{1} ∥$ and $∥ Σ_{m}^{- 1 / 2} ∥_{o p} \leq \sqrt{2 / λ}$ .

4.1F1F7step 3.1constructalgebra

Apply [F1] with $A = 3$ separately on each such block, obtaining iid standard Gaussian vectors $H_{m, j}$ , $1 \leq j \leq N_{m}$ . Put ${\tilde{G}}_{m, j} = Σ_{m}^{1 / 2} H_{m, j}$ . The operator norms of $Σ_{m}^{1 / 2}$ are uniformly bounded, so the maximum partial-sum discrepancy between the truncated row and this Gaussian row is at most $C N_{m}^{1 / a} (m + 1)^{3}$ , except with probability at most $C 2^{- 3 m}$ . Here $C$ is independent of the varying truncated law and $m$ , by the uniform constants in [F1].

4.2F2step 1.1step 3.1algebra

Set $D_{m} = Σ^{1 / 2} - Σ_{m}^{1 / 2}$ . If $A = Σ^{1 / 2}$ and $C_{m} = Σ_{m}^{1 / 2}$ , then $A D_{m} + D_{m} C_{m} = Σ - Σ_{m}$ . Taking Frobenius inner product with $D_{m}$ gives $(λ_{\min} (A) + λ_{\min} (C_{m})) ∥ D_{m} ∥_{F}^{2} \leq ∥ D_{m} ∥_{F} ∥ Σ - Σ_{m} ∥_{F}$ . Treating $D_{m} = 0$ separately and otherwise dividing, the eigenvalue bounds of step 3.1 and equivalence of these matrix norms in dimension at most two give $∥ D_{m} ∥_{o p} \leq C ∥ Σ - Σ_{m} ∥_{o p} \leq C L_{m}^{2 - q}$ . This uses positive definiteness only on $V$ , with no inverse on covariance-kernel directions.

5.1A1F3F4F5step 3.1step 4.1constructalgebra

To realize these couplings while preserving the original process, use [F4] on each finite coupling law to obtain a kernel giving its Gaussian block conditioned on its standardized truncated block. Evaluate that kernel at the actually given $(Z_{m, i})_{i \in I_{m}}$ . Apply [F5] with initial coordinate the entire original space $(Ω, F, P)$ and each subsequent coordinate its finite Gaussian block, the kernel depending on $ω$ through that block alone. This extension keeps the original marginal. Independence of the original disjoint blocks implies, by integrating finite block rectangles, that the unconditional paired-block laws factor into the finite coupling laws; thus all $H_{m, j}$ across all blocks are independent standard Gaussians. Assign independent Gaussian blocks to the finitely many early blocks before invertibility. Their total discrepancy is finite almost surely. On the resulting extension, [F3] and the summable bounds of step 4.1 give the stated partial-block coupling bound for all sufficiently large blocks almost surely.

6.1F3F6F7step 5.1step 4.2algebra

Put $G_{m, j} = Σ^{1 / 2} H_{m, j}$ . Each coordinate of $\sum_{j = 1}^{k} D_{m} H_{m, j}$ , $k \leq N_{m}$ , is Gaussian with variance at most $N_{m} ∥ D_{m} ∥_{o p}^{2}$ . By [F6], a union bound over the $d N_{m}$ coordinates and times bounds the probability that its maximum Euclidean norm exceeds $C_{1} ∥ D_{m} ∥_{o p} \sqrt{N_{m} (m + 1)}$ by $2 d 2^{m} \exp [- C_{1}^{2} (m + 1) / (2 d)]$ , summable in $m$ for fixed sufficiently large $C_{1}$ . If $D_{m} = 0$ the discrepancy is zero. Thus [F3] gives a maximum partial-block covariance-replacement error $O (N_{m}^{1 / 2} L_{m}^{2 - q} \sqrt{m + 1})$ almost surely. Its power $1 / 2 + (2 - q) / a$ is less than $1 / a$ because $a < q < 2 (q - 1)$ , so this error is $O (N_{m}^{1 / a} (m + 1)^{3})$ .

7.1F7step 1.1step 2.1step 5.1step 6.1algebra

Combine steps 2.1, 5.1 and 6.1 by triangle inequality. On one full event the maximum partial-block error between $X_{i}$ and $G_{m, j}$ is at most $C (ω) 2^{m / a} (m + 1)^{3}$ for every $m$ , after absorbing finitely many early errors into a finite random constant. Concatenate the independent Gaussian blocks into $(G_{i})$ . For $2^{J} \leq n < 2^{J + 1}$ , the global maximum is at most $C (ω) \sum_{m = 0}^{J} 2^{m / a} (m + 1)^{3} \leq C_{a} C (ω) 2^{J / a} (J + 1)^{3} = O (n^{1 / a} [\log (e n)]^{3})$ . Because $1 / a < 1 / p$ , this bound is $o (n^{1 / p})$ . Mapping from $V$ back to $R^{2}$ preserves the error norm and gives covariance equal to the original $Σ$ .

8.1F2F7F8F9step 1.1step 5.1step 7.1constructalgebra∎

To produce the Brownian driver, retain the independent standardized Gaussian increments in the effective eigenbasis of $Σ$ . If $d = 1$ , adjoin independent standard Gaussian increments in its orthogonal complement by [F7] and [F8]. For each coordinate and each integer interval, adjoin an independent Brownian motion $W_{i, l}$ by [F8], independent of all existing variables, and form its bridge $β_{i, l} (t) = W_{i, l} (t) - t W_{i, l} (1)$ , $0 \leq t \leq 1$ . The bridges are centered Gaussian with covariance $\min (s, t) - s t$ and zero endpoints. Interpolate each standardized Gaussian random walk on $[k, k + 1]$ by its linear increment plus $β_{k + 1, l}$ , independently for the two coordinates. By [F7], finite stacks are Gaussian; direct covariance calculation gives covariance $\min (s, t)$ within each coordinate and zero cross-coordinate covariance. Paths are continuous, including at integer joins. The covariance calculation makes vector increments over disjoint time intervals uncorrelated; finite joint Gaussianity and [F9] make them independent with covariance equal to interval length times $I_{2}$ , so [F9] identifies standard two-dimensional Brownian motion. Applying the deterministic orthogonal eigenbasis map makes a standard two-dimensional $B$ and yields $\sum_{i = 1}^{k} G_{i} = Σ^{1 / 2} B (k)$ for every integer $k$ . Step 7.1 supplies the asserted Brownian approximation and its weaker $O (n^{1 / p} \log n)$ rate.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Adjacent iid vector functionals admit a finite-moment strong approximation

Statement

Assume AC. Let $1 \leq d \leq 2$ , let $(E, E)$ be standard Borel, and let $(U_{i})_{i \geq 0}$ be iid $E$ -valued random elements with law $ν$ . Let $h : E^{2} \to R^{d}$ be measurable with $E h (U_{0}, U_{1}) = 0, E ∣ h (U_{0}, U_{1}) ∣^{q} < \infty$ for some finite $q > 2$ . Put $X_{i} = h (U_{i}, U_{i + 1})$ , and define $Γ = E [X_{0} X_{0}^{T}] + E [X_{0} X_{1}^{T}] + E [X_{1} X_{0}^{T}] .$ Then $Γ$ is positive semidefinite. For every fixed $2 < p < q$ , on an extension preserving the entire original process there are independent centered Gaussian vectors $G_{i}$ with covariance $Γ$ such that $\max_{1 \leq k \leq n} ∣ \sum_{i = 0}^{k - 1} (X_{i} - G_{i}) ∣ = o (n^{1 / p}) almost surely .$ The extension can also carry a standard $d$ -dimensional Brownian motion $B$ with $\sum_{i = 0}^{k - 1} G_{i} = Γ^{1 / 2} B (k)$ for every integer $k \geq 0$ . Singular $Γ$ , including $Γ = 0$ , is allowed. Neither the Gaussian row nor $B$ is required to be independent of the original process.

Facts & Assumptions

Given: AC, the standard-Borel innovation space, the iid sequence $U$ , the centered measurable $h$ with its finite $q$ th moment, and $2 < p < q$ . All vector norms are Euclidean and matrix inequalities are quadratic-form inequalities.

[A1]

AC supplies the inherited Gaussian, disintegration and countable-extension choices. (The Axiom of Choice)

[F1]

In dimension at most two, independent entries that are finite sums of independent centered atoms bounded by $B \geq 1$ , plus an independent Gaussian, and whose covariance matrices lie between $I$ and $K I$ , admit a uniform $C_{A, K} B \log^{3} (e N)$ Gaussian coupling with failure at most $N^{- A}$ for a row of length at most $N$ . The coupling is a measurable kernel in a supplied standard-Borel environment. (Heterogeneous bounded atom vector rows admit a logarithmic cubed Gaussian coupling)

[F2]

Independent centered vectors of dimension at most two, covariance $I$ and norm at most $R \geq 1$ , admit a Gaussian coupling of maximum partial-sum error $C_{A} R \log^{3} (e N)$ with failure at most $N^{- A}$ , uniformly over all row laws. An existing raw realization can be preserved. (Bounded independent vector rows admit a logarithmic cubed Gaussian coupling)

[F3]

Bounded independent centered real sums satisfy Bernstein's inequality with variance sum $v$ and atom bound $b$ : their two-sided tail at $t > 0$ is at most $2 \exp [- t^{2} / (2 (v + b t / 3))]$ , with zero sum when $v = 0$ . (Bernstein inequality for bounded independent sums)

[F4]

Conditional Jensen and Cauchy–Schwarz bound the moments and products below; Tonelli and Fubini allow the indicated nonnegative and absolutely integrable iterated expectations. (Conditional jensen inequality, Cauchy-Schwarz for random variables, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product)

[F5]

Real symmetric matrices have orthonormal eigenbases. Gaussian characteristic functions specify their covariances; independent Gaussian vectors form jointly Gaussian stacks, and uncorrelated jointly Gaussian blocks are independent. Standard-normal tails obey $\Pr (∣ Z ∣ > t) \leq 2 e^{- t^{2} / 2}$ for $t \geq 1$ . (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis, Characteristic function of a multivariate normal law, A finite stack of independent Gaussian vectors is jointly Gaussian, Uncorrelated jointly Gaussian blocks are independent, Two-sided Mills bounds for the standard normal tail)

[F6]

Standard-Borel spaces and their finite products admit disintegration of joint laws; finite Euclidean spaces are standard Borel. Kernel integrals and finite kernel products are measurable. Successive supplied kernels admit a countable extension preserving the initial probability space. (Standard Borel spaces, Finite products of standard Borel spaces are standard Borel, Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel, Disintegration of a joint law on standard borel spaces, Measurability of integration against a kernel, Finite products of probability kernels are probability kernels, Ionescu-Tulcea construction of a Markov chain)

[F7]

Markov's inequality and the finite union bound control exceptional events; summable exceptional probabilities occur only finitely often almost surely. (Markov's inequality for random variables, The finite union bound, The first Borel-Cantelli lemma for measures)

[F8]

Independent Brownian copies and countable products exist under AC; continuous centered Gaussian paths with the Brownian covariance and independent coordinate increments give standard vector Brownian motion. (Existence and scaling of $d$ -dimensional Brownian motion, Assuming countable and dependent choice, countable products of arbitrary probability spaces, $d$ -dimensional Brownian motion)

[F9]

Variance and covariance are centered square and product moments. (Moments, variance, and covariance on a probability space)

Proof

1.1F4F9givenconstructalgebra

Write $a (x) = \int h (x, y) ν (d y)$ , $b (y) = \int h (x, y) ν (d x)$ and $c (x, y) = h (x, y) - a (x) - b (y)$ , defining them arbitrarily on the null sets where the integrals are not finite. By [F4], they are measurable, have finite $q$ th moments, are centered, and $E [c (U, V) ∣ U] = E [c (U, V) ∣ V] = 0$ for independent $U, V$ of law $ν$ . Expanding the products using these two zero conditional means gives $Γ = cov (a (U) + b (U)) + cov (c (U, V)) \geq 0.$ Indeed the variance of $h (U, V)$ is the sum of the covariances of $a (U)$ , $b (V)$ and $c (U, V)$ , while the lag-one covariance is $E [b (U) a (U)^{T}]$ . This proves both the displayed identity and nonnegativity without any limit theorem.

1.2F2F5F7constructalgebra

We derive the finite-horizon iid bound needed for this changing row directly from [F2]. Suppose $T_{1}, \dots, T_{N}$ are iid centered vectors with $E ∣ T_{1} ∣^{q} \leq M$ , and covariance $Σ$ whose eigenvalues lie in a fixed $[λ, Λ]$ with $λ > 0$ . Put $R_{2} = N^{1 / β}$ and $T_{i}^{'} = T_{i} 1_{{∣ T_{i} ∣ \leq R_{2}}} - μ$ , $μ = E [T_{1} 1_{{∣ T_{1} ∣ \leq R_{2}}}]$ . The tail union probability is at most $M N R_{2}^{- q}$ , and on its complement the maximum sum correction is at most $M N R_{2}^{1 - q} = O (R_{2})$ . Also $Σ - Σ^{'} = E [T_{1} T_{1}^{T} 1_{{∣ T_{1} ∣ > R_{2}}}] + μ μ^{T}$ , so $∥ Σ - Σ^{'} ∥ \leq C_{M} R_{2}^{2 - q}$ . Uniformly for all sufficiently large $N$ the eigenvalues of $Σ^{'}$ are in $[λ / 2, C]$ . The whitened $T_{i}^{'}$ have covariance $I$ and norm at most $C R_{2}$ , so [F2] yields independent standard $H_{i}$ with maximum coupling error to $Σ^{' 1 / 2} H_{i}$ at most $C R_{2} \log^{3} (e N)$ , outside probability $N^{- A}$ . All constants are uniform in the iid law under these moment and eigenvalue bounds.

2.1F4F5F7F8step 1.1constructalgebra

For a deterministic orthonormal basis of $\ker Γ$ , step 1.1's sum of nonnegative scalar variances implies that each basis component of $c$ is zero almost surely and the same component of $a + b$ is zero almost surely. Hence the kernel projection of $h (U_{i}, U_{i + 1})$ is $a_{0} (U_{i}) - a_{0} (U_{i + 1})$ , where $a_{0}$ is the kernel projection of $a$ . It is an exact endpoint coboundary simultaneously for all $i$ , after removing a countable union of null sets. Its partial sums are $a_{0} (U_{0}) - a_{0} (U_{k})$ . For every $ϵ > 0$ , the union and Markov bounds over $k \leq 2^{j}$ give probability at most $C ϵ^{- q} 2^{j (1 - q / p)}$ that their maximum exceeds $ϵ 2^{j / p}$ . This is summable, so [F7], first for rational $ϵ > 0$ and then for all $n$ between successive powers of two, gives maximum $o (n^{1 / p})$ . If $Γ = 0$ take $G_{i} = 0$ and adjoin a Brownian motion by [F8]. Otherwise work in its deterministic positive-covariance range of dimension one or two and later restore this kernel projection.

3.1F4F5step 1.1step 2.1algebra

For independent anchors $A, B$ and interior variables $V, W$ , all of law $ν$ , put $Y = h (A, V) - a (A) + h (V, W) + h (W, B) - b (B)$ . Its conditional mean given $A, B$ is zero. Its average conditional covariance is $Γ_{*} : = E cov (Y ∣ A, B) = 2 cov (a (U) + b (U)) + 3 cov (c (U, V)) .$ To check this, substitute the decomposition from step 1.1 to write $Y = (a + b) (V) + (a + b) (W) + c (A, V) + c (V, W) + c (W, B)$ . Every cross product vanishes after integrating an endpoint using the zero conditional means of $c$ ; the five individual covariance contributions give the formula. Thus $2 Γ \leq Γ_{*} \leq 3 Γ$ , so $Γ_{*}$ is positive definite on the range retained in step 2.1. A fixed invertible linear map there makes $Γ_{*} = I$ ; its inverse will be restored at the end, and all subsequent constants may depend on this fixed map.

4.1F4F5step 3.1givenconstructchoosealgebra

Choose fixed $p < β < α < q$ , and consider an epoch of $N = 2^{j}$ consecutive triple indices. Put $R = N^{1 / α}$ , $L = K \log (e N)$ and $m = ⌈ R^{2} L ⌉$ . In this epoch replace the normalized active $h$ by $f = h 1_{{∣ h ∣ \leq R}} - E [h 1_{{∣ h ∣ \leq R}}]$ . Then $∣ f ∣ \leq 2 R$ , $E ∣ f ∣^{q} \leq C$ , and its means $a_{R}, b_{R}$ and degenerate component $c_{R}$ have uniformly bounded $q$ th moments by [F4]. They converge in $L^{2}$ to their untruncated versions: the squared tail is at most $R^{2 - q} E ∣ h ∣^{q}$ , and conditional averaging is an $L^{2}$ contraction by coordinatewise [F4]. Consequently the average conditional triple-noise covariance $Γ_{*, R} = 2 cov (a_{R} + b_{R}) + 3 cov (c_{R})$ tends to $I$ . For all sufficiently large deterministic epochs, $c_{0} I \leq Γ_{*, R} \leq C_{0} I$ for fixed $c_{0}, C_{0} > 0$ . Finitely many earlier epochs can be coupled arbitrarily and contribute only finite errors.

5.1F4F6F9step 4.1constructalgebra

Let $A_{t} = U_{3 t}$ , $V_{t} = U_{3 t + 1}$ and $W_{t} = U_{3 t + 2}$ in the epoch, relabeling its first triple index as zero. Conditional on its anchors $(A_{0}, \dots, A_{N})$ , the interior pairs $(V_{t}, W_{t})$ are independent, and the centered triple noises $Y_{t} = f (A_{t}, V_{t}) - a_{R} (A_{t}) + f (V_{t}, W_{t}) + f (W_{t}, A_{t + 1}) - b_{R} (A_{t + 1})$ are independent centered vectors bounded by $C R$ . Define the symmetric matrices $Q (z) = cov (f (V, W) + f (W, z) - b_{R} (z))$ and $P (z) = cov (f (z, V) - a_{R} (z)) + cov (f (z, V), f (V, W)) + cov (f (V, W), f (z, V))$ , where these covariances integrate independent $V, W$ . The left and right endpoint terms are independent given the anchors, so $V_{t} : = cov (Y_{t} ∣ A_{t}, A_{t + 1}) = P (A_{t}) + Q (A_{t + 1}), D_{t} : = P (A_{t}) + Q (A_{t}) = cov (Y (A_{t}, A_{t}) ∣ A_{t}) .$ Although $P$ need not be positive semidefinite, $V_{t}, D_{t}, Q (A_{t})$ are positive semidefinite, their norms are at most $C R^{2}$ , and $E V_{t} = E D_{t} = Γ_{*, R}$ . Moreover, for every interval of triple indices, $\sum_{t = u}^{v - 1} (V_{t} - D_{t}) = Q (A_{v}) - Q (A_{u}) .$ These are identities of measurable anchor functions, by [F4] and [F6].

6.1F3F5F7step 4.1step 5.1choosealgebra

Divide the epoch into consecutive chunks of length $m$ , leaving a shorter final chunk. The entries of $V_{t}$ are one-dependent functions of adjacent iid anchors, while $D_{t}$ are iid functions of individual anchors. For a positive semidefinite matrix $T$ of norm at most $C R^{2}$ , $∣ T_{u v} ∣^{2} \leq T_{u u} T_{v v} \leq C R^{2} T_{u u}$ . Its entrywise second moments here are therefore at most $C R^{2}$ , because the mean matrices in step 5.1 are uniformly bounded. Within each parity the entries of $V_{t}$ are independent. Apply [F3] separately to both parities and each matrix entry, and to the independent entries of $D_{t}$ across $t$ . For any full chunk $I$ , the probability that an entry of either centered covariance sum exceeds a fixed small multiple of $m$ is at most $C \exp (- c m / R^{2})$ . Since $∥ T ∥_{o p} \leq d \max_{u, v} ∣ T_{u v} ∣$ for symmetric matrices in dimension $d \leq 2$ , choose the multiplier small relative to $c_{0}$ and union over at most $N$ chunks and their entries. Choosing $K$ sufficiently large gives an anchor event of probability at least $1 - N^{- A}$ , for any prescribed $A > 0$ , on which $c_{1} m I \leq \sum_{t \in I} V_{t} \leq C_{1} m I, c_{1} m I \leq \sum_{t \in I} D_{t} \leq C_{1} m I$ for every full chunk, with both sums bounded on both sides. The same entrywise Bernstein calculation gives an upper bound $C_{1} m I$ on both sums in the shorter chunk. Here and below the constants may be enlarged independently of $N$ and $R$ . Since $α > 2$ , $m < N$ eventually; if not, the same upper bound handles the entire epoch as one residual chunk.

7.1A1F1F6step 5.1step 6.1constructalgebra

On a good anchor vector, divide each full-chunk sum of $Y_{t}$ by $\sqrt{c_{1} m}$ . Its conditional covariance lies between $I$ and $K I$ , for a fixed $K$ , and it is a sum of independent centered atoms of norm at most $C R / \sqrt{m} \leq C$ . Its whole value need not be bounded. Its atom kernels are the measurable pushforwards of $ν \otimes ν$ under the formula in step 5.1; independence holds conditional on the supplied anchors. Thus [F1], with the ambient length parameter $N$ and a fixed atom bound $B \geq \max (1, C)$ , supplies a measurable coupling of the chunk sums with independent matching Gaussian chunk sums. Undoing the normalization gives maximum chunk-boundary error at most $C \sqrt{m} \log^{3} (e N) \leq C R \log^{7 / 2} (e N)$ , with conditional failure at most $N^{- A}$ . On bad anchors use independent product coupling instead, preserving both marginals. No theorem about uniformly bounded grouped values is being invoked.

7.2F5step 5.1step 6.1constructalgebra

To redistribute covariance without a matrix random clock, start instead with independent standard Gaussian vectors $Z_{t}$ , independent of the anchors, and put $Z_{t}^{D} = D_{t}^{1 / 2} Z_{t}$ . These have the required conditional independent $D_{t}$ row law even if individual $D_{t}$ are singular. On good anchors group them into the same full chunks, whose covariances $B_{i} = \sum D_{t}$ are positive definite. Write $A_{i} = \sum V_{t}$ for the desired other row. Pad to a dyadic number of chunks with independent identical-covariance $m I$ Gaussian entries in both rows. All chunk covariances lie between $c_{1} m I$ and $C_{1} m I$ , and every contiguous whole-chunk interval satisfies $∥ A_{I} - B_{I} ∥ \leq C R^{2}$ by step 5.1's endpoint identity; padded intervals retain that bound. We construct the $A$ row from the already given $B$ row using the Gaussian tree in the next two steps, leaving the original $Z_{t}$ unchanged.

8.1F3F5F6F7step 6.1step 7.1constructalgebra

Fill a Gaussian chunk of total covariance $S > 0$ and coupled sum $z$ with increments of individual covariances $V_{t}$ by taking independent $H_{t} \sim N (0, V_{t})$ and setting $H_{t}^{'} = H_{t} + V_{t} S^{- 1} (z - \sum H_{t})$ . The residual $H_{t} - V_{t} S^{- 1} \sum H_{t}$ is independent of the original total Gaussian by [F5]; direct covariance calculation shows this is its exact conditional bridge law and its sum is $z$ . Thus the completed Gaussian noise row has exactly its conditional independent Gaussian product law. Conditional on good anchors, every within-chunk raw prefix has atom bound $C R$ and covariance at most $C_{1} m I$ . Coordinatewise [F3] and Gaussian tails in [F5], with a union bound over at most $N$ prefixes, bound both raw and Gaussian within-chunk norms by $C (\sqrt{m L} + R L) \leq C R L$ , with failure at most $N^{- A}$ after enlarging $K$ . Couple a shorter residual chunk independently and use its covariance upper bound in the same calculation. Consequently there is a conditional noise-row coupling $Y_{t} \leftrightarrow Z_{t}^{V}$ , with $Z_{t}^{V}$ conditionally independent $N (0, V_{t})$ , of full-prefix error $C R \log^{7 / 2} (e N)$ and total failure $C N^{- A}$ . Its extension to the actually given $Y_{t}$ follows by [F6] disintegrating the finite joint law given anchors and $Y_{t}$ ; the original innovations are preserved.

8.2F5F7step 7.2constructalgebra

At a node of $s$ chunks and for either covariance row $C$ , write $C_{P} = C_{L} + C_{R}$ , $H_{C} = C_{L} C_{P}^{- 1}$ , and $T_{C} = C_{L} - C_{L} C_{P}^{- 1} C_{L} = (C_{L}^{- 1} + C_{R}^{- 1})^{- 1}$ . Thus $C_{P}$ and $T_{C}$ have eigenvalues comparable to $s m$ , and the Gaussian parent and residual $S_{L} - H_{C} S_{P}$ are independent by [F5]. Extract their standardized independent Gaussian primitives from the given $B$ row, and use the same primitives to construct the $A$ root and residuals. For positive definite matrices $M, N$ , the identity $M^{1 / 2} X + X N^{1 / 2} = M - N$ , with $X = M^{1 / 2} - N^{1 / 2}$ , yields $∥ X ∥ \leq C ∥ M - N ∥ / (\sqrt{λ_{\min} M} + \sqrt{λ_{\min} N})$ : solve it by integrating $e^{- t M^{1 / 2}} (M - N) e^{- t N^{1 / 2}}$ and bound the integral. The endpoint bounds in step 7.2 and the inverse identity imply $∥ H_{A} - H_{B} ∥ \leq C R^{2} / (s m)$ , $∥ T_{A} - T_{B} ∥ \leq C R^{2}$ , and $∥ T_{A}^{1 / 2} - T_{B}^{1 / 2} ∥ \leq C R^{2} / \sqrt{s m}$ . A union bound over the Gaussian parents and primitives gives, outside probability $N^{- A}$ , $∣ S_{P}^{B} ∣ \leq C \sqrt{s m L}$ and primitive norms at most $C \sqrt{L}$ . Reconstruct children by $S_{L}^{C} = H_{C} S_{P}^{C} + T_{C}^{1 / 2} z_{I}$ and $S_{R}^{C} = S_{P}^{C} - S_{L}^{C}$ . Their errors satisfy $e_{L} = H_{A} e_{P} + η_{I}$ , $e_{R} = (I - H_{A}) e_{P} - η_{I}$ , with $∣ η_{I} ∣ \leq C R^{2} \sqrt{L / (s m)} \leq C R$ , and the root error is also at most $C R$ . Every child and root marginal is its exact Gaussian law, so the two completed chunk rows each have their required conditional product law.

9.1F5F6F7step 6.1step 8.1step 7.2step 8.2algebra

The prefix bound uses exact covariance interpolation, not repeated norm bounds on $H_{A}$ . In any subtree its leading root coefficient is $A_{p r e f i x} A_{P}^{- 1}$ . At a boundary ancestor its residual coefficient is $A_{p a r t i a l l e f t} A_{L}^{- 1}$ if the boundary is in the left child, and $I - A_{p a r t i a l r i g h t} A_{R}^{- 1}$ if it is in the right child. For the left case substitute the child's leading term and cancel $A_{L}^{- 1} A_{L}$ ; for the right case add the full left error to the partial right error, giving leading coefficient $(A_{L} + A_{p a r t i a l r i g h t}) A_{P}^{- 1}$ and the stated residual coefficient. This proves the expansion by induction. Each coefficient has norm at most $1 + C_{1} / c_{1}$ , using the count and eigenvalue bounds on chunk covariances, and there are at most $C \log (e N)$ boundary ancestors. Step 8.2 therefore gives prefix error $C R \log (e N)$ at chunk boundaries. Fill the $V_{t}$ Gaussian increments by the bridge in step 8.1; the $D_{t}$ increments were already present. Their within-chunk prefix norms are at most $C \sqrt{m L} \leq C R L$ by Gaussian tails and the covariance upper bounds. Handle a short residual chunk independently. This gives full-prefix Gaussian-row error $C R \log (e N)$ outside probability $C N^{- A}$ . On bad anchors use an independent $V_{t}$ row while retaining the original independent standard $Z_{t}$ . All operations are measurable matrix functions and finite Gaussian kernels.

10.1F4F6F7step 4.1step 8.1step 9.1constructalgebra

The couplings of steps 8.1 and 9.1 have the same joint anchor and $V_{t}$ -Gaussian-row marginal. Disintegrate the second joint law to give its standard $Z$ row conditional on that common anchor and Gaussian row, and append it to the first coupling. The disintegration formula in [F6] proves that both joint laws, hence both error estimates, are preserved. In particular the original innovations remain unchanged and the $Z_{t}$ row is still independent standard Gaussian conditional on anchors. The two full-prefix errors add. Adding back the conditional means of the triples now gives an iid replacement row $L_{t} = D_{t}^{1 / 2} Z_{t} + a_{R} (A_{t}) + b_{R} (A_{t}),$ since the pairs $(A_{t}, Z_{t})$ are iid. Indeed the original mean sum is $\sum_{t < k} (a_{R} (A_{t}) + b_{R} (A_{t + 1})) = \sum_{t < k} (a_{R} (A_{t}) + b_{R} (A_{t})) + b_{R} (A_{k}) - b_{R} (A_{0})$ . The endpoint means have uniformly bounded $q$ th moments by step 4.1. Their maximum exceeds $R$ with probability at most $C N R^{- q}$ by [F7]. Therefore the original truncated triple sums and the partial sums of $L_{t}$ have maximum error $C R \log^{7 / 2} (e N)$ with failure at most $C N^{- A} + C N R^{- q}$ .

11.1F4F5F9step 1.1step 2.1step 4.1step 5.1step 10.1algebra

The row $L_{t}$ is centered and has uniformly bounded $q$ th moment: conditional Gaussian moments and coordinatewise [F4] give $E ∥ D_{t} ∥^{q / 2} \leq C E ∣ Y (A_{t}, A_{t}) ∣^{q} \leq C E ∣ f (U, V) ∣^{q} \leq C$ , and the mean terms satisfy the same bound. Write $γ_{0} = E [f (U_{0}, U_{1}) f (U_{0}, U_{1})^{T}]$ , $γ_{1} = E [f (U_{0}, U_{1}) f (U_{1}, U_{2})^{T}]$ , and $Γ_{R} = γ_{0} + γ_{1} + γ_{1}^{T}$ . The variance of a triple is $3 γ_{0} + 2 (γ_{1} + γ_{1}^{T})$ . Subtracting the covariance of its conditional mean $a_{R} (A_{0}) + b_{R} (A_{1})$ gives $E D_{t}$ by total variance. Since $E [b_{R} (U) a_{R} (U)^{T}] = γ_{1}$ , adding $cov (a_{R} + b_{R})$ gives $cov (L_{t}) = 3 Γ_{R} .$ This is a direct finite product calculation, independent of convergence in law or an inference from a coupling error. As $R \to \infty$ , $Γ_{R} \to Γ$ , which is positive definite in the active space; hence these iid covariances have fixed upper and positive lower eigenvalue bounds in every sufficiently large epoch.

12.1F5F7step 8.2step 11.1step 1.2algebra

Step 8.2's square-root bound gives $∥ Σ^{1 / 2} - Σ^{' 1 / 2} ∥ \leq C_{M} R_{2}^{2 - q}$ . Use the same $H_{i}$ and apply Gaussian tails to all its partial sums to bound the covariance correction by $C_{M} \sqrt{N} R_{2}^{2 - q} \sqrt{\log (e N)}$ with failure at most $N^{- A}$ . This is $o (R_{2})$ because $β < q < 2 (q - 1)$ . Consequently step 1.2 and this covariance correction give a finite-row iid coupling with matching covariance $Σ$ , error $C R_{2} \log^{3} (e N)$ and failure $C N^{- A} + M N R_{2}^{- q}$ . Apply this result to the varying laws $T_{i} = L_{i}$ from step 11.1, using their uniform moment and eigenvalue bounds. This uses no fixed-law asymptotic iid theorem and asserts no triangular-law interface absent from a supplier's Statement.

13.1F4F5F7step 4.1step 8.2step 10.1step 11.1step 12.1algebra

The original normalized active $h$ and $f$ agree up to their global mean correction unless one of the epoch's $3 N$ observations exceeds $R$ , an event of probability at most $C N R^{- q}$ . The mean correction is at most $C N R^{1 - q} = O (R)$ . Moreover $∥ Γ_{R} - Γ ∥ \leq C R^{2 - q}$ . For the lag-one product difference, use $∣ h_{0} h_{1}^{T} - f_{0} f_{1}^{T} ∣ \leq C (∣ h_{0} ∣ + ∣ h_{1} ∣)^{2} 1_{{∣ h_{0} ∣ + ∣ h_{1} ∣ > R}}$ before centering and then include the mean correction. Pointwise the displayed squared tail is at most a constant times the sum of the squared tails of $∣ h_{0} ∣$ and $∣ h_{1} ∣$ above $R / 2$ , so its expectation is at most $C R^{2 - q} E ∣ h ∣^{q}$ . The variance term has the same bound. Since $Γ$ is positive definite in the active space, couple the iid Gaussian triple increments of covariance $3 Γ_{R}$ from step 12.1 to increments of covariance $3 Γ$ using the same standard normals. Step 8.2's square-root bound and Gaussian tails make the maximum correction at most $C \sqrt{N} R^{2 - q} \sqrt{\log (e N)} = o (R)$ with failure $N^{- A}$ , because $α < q < 2 (q - 1)$ . Thus each sufficiently large epoch of original triple sums can be coupled to independent Gaussian increments of covariance $3 Γ$ , with error $C N^{1 / β} \log^{7 / 2} (e N)$ and failure at most $C N^{- A} + C N^{1 - q / α} + C N^{1 - q / β} .$ Here $R = N^{1 / α} \leq N^{1 / β}$ , and all constants are uniform over epochs.

14.1A1F6F7step 5.1step 8.1step 7.2step 10.1step 12.1step 13.1constructalgebra

Retain triple index $t = 0$ as a finite initial prefix, coupled arbitrarily, and perform the finite constructions on epochs ${2^{j}, \dots, 2^{j + 1} - 1}$ of triple indices, $j \geq 0$ , using fresh kernels. Each conditional noise coupling and each Gaussian redistribution has the full conditional product marginal given all anchors. Because the original interior pairs are independent conditional on anchors, their finite-epoch kernel products give a product standard $Z$ law conditional on all anchors. Hence the pairs $(A_{t}, Z_{t})$ , and therefore the replacement variables $L_{t}$ with the deterministic epoch-dependent function in step 10.1, are independent across epochs as well as within them. Each later iid-row coupling is glued through its finite replacement row by [F6]; because these rows are independent, products of their kernels give independent final Gaussian rows across epochs. These statements follow by integrating finite rectangles under the conditional product laws, so no independence is being inferred from an error bound. All anchor sets and row coordinates involved in an epoch are finite standard-Borel products, making these disintegrations legitimate. Apply [F6]'s countable kernel extension with the entire original probability space as its initial coordinate to preserve all original innovations. The failure bounds in step 13.1 are summable over $N = 2^{j}$ , since $β < α < q$ . By [F7], all sufficiently large epochs satisfy their error bounds almost surely; finitely many earlier errors can be included in a finite random constant.

15.1F5F7step 2.1step 3.1step 13.1step 14.1constructalgebra

Sum the completed epoch errors and the current partial one. The geometric bound $\sum_{j \leq J} 2^{j / β} (j + 1)^{7 / 2} \leq C_{β} 2^{J / β} (J + 1)^{7 / 2}$ gives a global triple-prefix error $O (n^{1 / β} \log^{7 / 2} (e n)) = o (n^{1 / p})$ . Map back from the fixed normalization in step 3.1 and restore the covariance-kernel coboundary from step 2.1. Split each independent Gaussian triple increment $H_{t} \sim N (0, 3 Γ)$ into three increments of covariance $Γ$ : for each $t$ take a fresh triple $K_{t, 0}, K_{t, 1}, K_{t, 2} \sim N (0, Γ)$ , independent across all $t, r$ and of all triple increments, and put $G_{3 t + r} = H_{t} / 3 + K_{t, r} - (K_{t, 0} + K_{t, 1} + K_{t, 2}) / 3$ . By [F5] these three increments are jointly Gaussian, each has covariance $Γ$ and distinct ones have zero cross covariance, so the entire resulting row is independent with covariance $Γ$ . At nontriple endpoints the original sum changes by at most two $X_{i}$ , and the Gaussian sum by at most two $G_{i}$ . Union and Markov bounds on dyadic ranges, using respectively their finite $q$ th moments and arbitrary Gaussian moments, show both maxima are $o (n^{1 / p})$ , as in step 2.1. This proves the stated full-prefix approximation.

16.1A1F5F8step 2.1step 15.1constructalgebra∎

In a fixed orthonormal eigenbasis of $Γ$ , divide each positive-variance Gaussian coordinate by its fixed standard deviation and add independent standard Gaussian coordinates in the zero-variance directions. This yields independent standard normal vector increments realizing $G_{i}$ by $Γ^{1 / 2}$ . For each integer interval adjoin independent Brownian copies by [F8] and use their bridges $W (t) - t W (1)$ , independently of the existing increments. Interpolate the standard Gaussian random walk linearly on that interval plus its bridge. The paths are continuous and direct covariance calculation gives $\min (s, t) I_{d}$ ; finite Gaussianity and zero cross covariances give independent normal increments by [F5]. Thus [F8] identifies a standard $d$ -dimensional Brownian motion, and the fixed eigenbasis map gives $\sum_{i < k} G_{i} = Γ^{1 / 2} B (k)$ at every integer $k$ . This construction also applies when $Γ = 0$ by adjoining an independent Brownian motion. It does not alter the original-process marginal or any error already proved.

Source notes

The conditional triple decomposition follows the finite-memory mechanism in Berkes–Liu–Wu. Every coupling theorem used as a premise is proved locally. The first coupling uses independent bounded atoms in a random environment, the covariance redistribution is a direct Gaussian matrix tree, and the changing iid replacement laws are handled by the explicit finite-horizon argument in steps 1.2 and 12.1. No independent-nonidentical strong approximation, matrix random clock, or uniform triangular-law asymptotic theorem is imported.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Strong invariance principle for geometrically ergodic chains

Statement

Let $P$ be a geometrically ergodic Markov chain with invariant probability $π$ , let $g$ be real Borel, and suppose there are $δ, ε > 0$ with $π (∣ g ∣^{2 + δ + ε}) < \infty$ . Suppose also that a state-dependent one-step minorization State-dependent one-step minorization holds. For the Flegal–Jones strong invariance principle convention, assume its asymptotic standard deviation satisfies $0 < σ_{g} < \infty$ . Then on a suitable extension of the chain's probability space there are a standard Brownian motion $B$ , a finite random constant $C (ω)$ , and a finite random index $N (ω)$ such that for all $n \geq N (ω)$ , $∣ \sum_{i = 1}^{n} (g (X_{i}) - π (g)) - σ_{g} B (n) ∣ \leq C (ω) n^{1 / (2 + δ)} \log n$ almost surely.

Facts & Assumptions

Given: The chain, $g$ , $δ, ε$ and positive finite $σ_{g}$ in the Statement. AC is assumed under this item's axiom-use convention. The initial law is arbitrary.

[A1]

AC is the convention for the Gaussian, path-law, standard-Borel disintegration and probability-space extension suppliers below. The Axiom of Choice

[F1]

For each specified initial law, an everywhere geometrically ergodic kernel on an arbitrary measurable space has a standard-Borel observation factor with the same initial and stationary real-observation path laws and moments. A joint observation/standard-Borel auxiliary law lifts to an extension of the entire original probability space. A geometrically ergodic chain has an all-start standard-Borel observation factor

[F2]

On a standard-Borel geometrically ergodic chain, a finite centered observable with stationary $v$ th moment has, after an almost surely finite integer delay, actual adjacent-iid full calendar tours: lengths $L_{j} = L (U_{j}) \geq 1$ are iid with exponential moment, rewards $Z_{j} = z (U_{j}, U_{j + 1})$ are centered, and absolute tour envelopes $H_{j} = h (U_{j}, U_{j + 1})$ have every moment $1 \leq q < v$ . The full-tour occupation identity is $E \sum b (X_{t}) = μ_{L} π (b)$ , where $μ_{L} = E L_{0}$ . The same fixed post-delay tour law can be used for the specified and invariant initial laws. Geometric Markov chains admit delayed adjacent-iid split tours with spare moments

[F3]

Centered adjacent-iid vectors of dimension at most two with finite $q$ th moment admit, for every $2 < p < q$ , an original-process-preserving Gaussian partial-sum approximation with uniform error $o (n^{1 / p})$ . Its long-run covariance is $Γ = E X_{0} X_{0}^{T} + E X_{0} X_{1}^{T} + E X_{1} X_{0}^{T}$ , and a standard vector Brownian driver is supplied, including singular covariance. Adjacent iid vector functionals admit a finite-moment strong approximation

[F4]

Centered iid vectors in dimension two with finite $q$ th moment, for $2 < p < a < q$ , have a preserving-extension Gaussian approximation of maximum error $O (n^{1 / a} \log^{3} (e n)) = o (n^{1 / p})$ , with a standard vector Brownian driver. Singular covariance is allowed, so scalar variables can be embedded as $(Y, 0)$ . Finite-moment iid vectors admit a strong Gaussian approximation

[F5]

If a rate- $λ$ Poisson clock lies in a sigma-algebra independent of a standard Brownian motion $B$ , there is an extended standard Brownian motion $W^{*}$ independent of that whole sigma-algebra such that $\sup_{t \leq n} ∣ B (t) - λ^{- 1 / 2} W^{*} (N (t)) ∣ = O (\log^{2} (n + 2))$ almost surely. Independent Brownian motion straightens a Poisson clock

[F6]

Centered identically distributed independent or one-dependent $L^{q}$ variables, for $q > p > 2$ , have uniform $o (n^{1 / p})$ increments over all windows of length at most $A n^{1 / p} (\log n)^{d}$ , simultaneously for fixed $A > 0, d \geq 0$ . Finite moments control increments over short moving windows

[F7]

A finite-window homogeneous Poisson process has, conditional on its total count, independent uniform unordered event positions; its ordered event density with $m$ total points in $(0, T]$ is $e^{- λ T} λ^{m}$ . The scalar Poisson mass and the convergent exponential series are available. Finite-window homogeneous Poisson construction and ordered event-time density, The scalar Poisson count law, The real exponential function and the number $e$ by a power series, The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$

[F8]

Real sequences are standard Borel through their countable Polish product presentation. Standard-Borel joint laws have conditional probability kernels, and supplied kernels extend an arbitrary original probability space without changing its marginal. The real line with its Borel sigma-algebra is standard Borel, Under the Axiom of Countable Choice, countable products and $G_{δ}$ subspaces of Polish spaces are Polish, Disintegration of a joint law on standard borel spaces, Ionescu-Tulcea construction of a Markov chain

[F9]

Markov's inequality, countable subadditivity and the first Borel–Cantelli lemma give the usual union and summability bounds. Markov's inequality for random variables, Finite and countable subadditivity of measures, First Borel-Cantelli lemma for events

[F10]

Brownian motion has continuous centered Gaussian paths with covariance $\min (s, t)$ . Uncorrelated jointly Gaussian blocks are independent. Independent Brownian copies exist. Reflection and the normal Mills upper bound give $P (\sup_{s \leq h} ∣ B (s) ∣ > x) \leq 4 e^{- x^{2} / (2 h)}$ when $x / \sqrt{h} \geq 1$ . Brownian motion, Uncorrelated jointly Gaussian blocks are independent, Existence of continuous Brownian motion, Assuming countable and dependent choice, countable products of arbitrary probability spaces, Law of the Brownian maximum, Two-sided Mills bounds for the standard normal tail

[F11]

An everywhere finite common geometric total-variation envelope admits an integrable geometric envelope, which implies exponential stationary strong mixing. A measurable observation does not increase those coefficients. Harris geometric ergodicity admits an integrable total-variation envelope, A total-variation rate bounds strong-mixing coefficients, A measurable observation does not increase stationary mixing coefficients

[F12]

A centered stationary $L^{2 + η}$ sequence with $\sum_{n} α (n)^{η / (2 + η)} < \infty$ has an absolutely convergent covariance sum and a CLT with that variance, including zero. A perturbation tending to zero in probability preserves a distributional limit; centered normal characteristic functions identify their variances. Strong-mixing CLTs under bounded, moment, and logarithmic-moment conditions, Converging together lemma, Characteristic function of a normal law, Long-run variance for a stationary Markov chain

Proof

1.1A1F1F2givenchooseconstructalgebra

Put $P_{0} = 2 + δ$ , $Q_{0} = 2 + δ + ε$ , and $f = g - π g$ . The given moment makes $π ∣ g ∣ < \infty$ , since $∣ g ∣ \leq 1 + ∣ g ∣^{Q_{0}}$ , and the finite-sum power inequality gives $π ∣ f ∣^{Q_{0}} < \infty$ . Choose $P_{0} < p < q < Q_{0}$ and set $b = 1 / p < 1 / P_{0} < 1 / 2$ . To obtain exactly the requested observation indices, apply [F1] to the chain starting at $X_{1}$ , with its actual initial law, and observable $f$ . This gives a standard-Borel factor retaining the full path $(f (X_{1}), f (X_{2}), \dots)$ , the stationary path and its moments, and an everywhere finite common geometric bound. Work on this factor until the original-space lift at the end. Apply [F2] there, retaining its finite prefix and integer delay $D$ . Define $F (t)$ to be the sum of the first $⌊ t ⌋$ centered observations of the retained tail beginning at calendar position $D$ , with the empty sum zero. This real-time tail process has cycle lengths $L_{j}$ , centered rewards $Z_{j}$ , envelopes $H_{j}$ , and relative boundaries $T_{0} = 0$ , $T_{k} = \sum_{j < k} L_{j}$ . Then $F (T_{k}) = \sum_{j < k} Z_{j}$ , $E H_{0}^{q} < \infty$ , and $μ = E L_{0} \in [1, \infty)$ . The pair $(Z_{j}, L_{j} - μ)$ is a centered adjacent-iid vector in $L^{q}$ ; exponential length moments and $∣ Z_{j} ∣ \leq H_{j}$ verify its two coordinate moments. No iid assertion about the full tours is used.

2.1F2F3F10step 1.1constructalgebra

Apply [F3] jointly to $(Z_{j}, L_{j} - μ)$ . Write its long-run covariance as $Γ = (\begin{array}{cc} u & c \\ c & s^{2} \end{array}) \geq 0$ . Since the lengths are iid, $s^{2} = Var (L_{0})$ ; the reward and cross entries still include their actual lag-one terms. The Gaussian approximation gives a continuous centered Gaussian vector process $A$ of covariance $\min (t, r) Γ$ such that $\max_{k \leq n} ∣ F (T_{k}) - A_{1} (k) ∣ = o (n^{b})$ and $\max_{k \leq n} ∣ T_{k} - μ k - A_{2} (k) ∣ = o (n^{b})$ . For now suppose $s > 0$ . Put $a = c / s^{2}$ and $v^{2} = u - c^{2} / s^{2} \geq 0$ . Then $B_{0} = A_{2} / s$ is standard Brownian. If $v > 0$ , $B = (A_{1} - a A_{2}) / v$ is standard Brownian independent of the entire process $B_{0}$ : each finite stack is jointly Gaussian and every cross-covariance is $(c - a s^{2}) \min (t, r) = 0$ , so [F10] gives independence on finite rational-time cylinders and continuity extends it to all times. If $v = 0$ , the residual Gaussian process is zero on rational times almost surely and hence identically zero; adjoin an independent unused $B$ . Thus, with independent $B, B_{0}$ , we have $\max_{k \leq n} ∣ F (T_{k}) - a (T_{k} - μ k) - v B (k) ∣ = o (n^{b})$ and $\max_{k \leq n} ∣ T_{k} - μ k - s B_{0} (k) ∣ = o (n^{b})$ . This regression uses the full $c = Γ_{12}$ , not just the covariance inside a single cycle.

3.1A1F4F7F8F10step 2.1chooseconstructalgebra

Define $γ = s^{2} / μ$ and $λ = μ^{2} / s^{2}$ , so $λ γ = μ$ and $γ \sqrt{λ} = s$ . Choose any fixed $r > p$ and $r < a_{1} < q_{1} < \infty$ . Apply [F4] to the centered iid scalar Poisson variables $C_{i} - λ$ , embedded as $(C_{i} - λ, 0)$ , obtaining a joint law with a standard Brownian driver $B_{0}^{'}$ and $\max_{k \leq n} ∣ \sum_{i = 1}^{k} C_{i} - λ k - \sqrt{λ} B_{0}^{'} (k) ∣ = o (n^{1 / r})$ . Their moments of every order follow directly from the Poisson exponential series in [F7]. By [F8], disintegrate this joint law to a kernel for the entire count sequence given the integer driver sequence, and evaluate it at the already present $(B_{0} (k))_{k \geq 0}$ . This preserves all original variables and reproduces the asserted count/driver joint law. The kernel depends only on $B_{0}$ and is sampled using new randomness independent of everything earlier, so $B$ remains independent of the joint pair $(B_{0}, (C_{i})_{i})$ . Conditional on $C_{i} = m$ , insert $m$ independent uniform positions in $(i - 1, i]$ , independently over unit intervals and of earlier variables given the counts. By [F7] these are the exact finite-window Poisson laws. More explicitly, partitioning a unit interval into lengths $d_{j}$ gives multinomial conditional counts; summing their mass against $Pois (λ)$ yields the product $\prod_{j} e^{- λ d_{j}} (λ d_{j})^{k_{j}} / k_{j}!$ . Hence the resulting locally finite simple right-continuous clock $N$ has independent Poisson increments of rate $λ$ , $N (k) = \sum_{i \leq k} C_{i}$ , and remains independent of $B$ jointly with $B_{0}$ .

3.2F2F6F9F10step 1.1step 2.1algebra

Two uniform bounds control the remaining calendar errors. By the common $q$ th envelope moment and [F9], $P (\max_{j \leq C 2^{h}} H_{j} > η 2^{h / p}) \leq C^{'} η^{- q} 2^{h (1 - q / p)}$ , summable in $h$ . Intersecting Borel–Cantelli events over positive rational $η$ and integer $C$ yields $\max_{j \leq C n} H_{j} = o (n^{b})$ without any independence requirement. For any standard Brownian motion and fixed $A, C > 0$ , cover $[0, C 2^{h}]$ by overlapping intervals of length $2 A \cdot 2^{h b}$ . Every pair of times at distance at most $A \cdot 2^{h b}$ lies in one such interval, and [F10] bounds the oscillation tail at $K \sqrt{2^{h b} h}$ by $4 e^{- c K^{2} h}$ . There are at most $C^{'} 2^{h (1 - b)} + 2$ intervals. Taking $K$ large and using [F9] gives the uniform modulus $O (n^{b / 2} \sqrt{\log n}) = o (n^{b})$ for windows of length $A n^{b}$ ; for fixed window length the same proof gives $O (\sqrt{\log n})$ . All fixed integer $A, C$ can be placed on one event. Finally [F6] applies to the centered one-dependent rewards $Z_{j}$ , so their partial-sum increments over at most $A n^{b}$ consecutive cycle indices, among the first $C n$ indices, are uniformly $o (n^{b})$ . A finite random bound for $A$ or $C$ is covered by their countably many integer choices.

4.1F7F9F10step 2.1step 3.1constructalgebra

We record the uniform estimates used for this clock and for Brownian motion. Summing the Poisson mass series gives $E e^{θ (N (v) - N (u) - λ (v - u))} = \exp (λ (v - u) (e^{θ} - 1 - θ))$ . The exponential series and $k! \geq 2 \cdot 3^{k - 2}$ for $k \geq 2$ give $e^{θ} - 1 - θ \leq θ^{2} / [2 (1 - ∣ θ ∣ / 3)]$ for $∣ θ ∣ < 3$ . Exponential Markov with parameter $x / (λ (v - u) + x / 3)$ and either sign therefore bounds the increment deviation tail by $2 \exp [- c_{λ} \min {x^{2} / (v - u), x}]$ . On dyadic horizons $n$ , take $x = K (\sqrt{(v - u) \log (n + 2)} + \log (n + 2))$ and union over the at most $(C n + 1)^{2}$ integer intervals in $[0, C n]$ . A sufficiently large deterministic $K$ makes the probabilities summable, so [F9] proves this bound for all such intervals eventually almost surely, for every positive integer $C$ by countable intersection. In particular unit counts are $O (\log n)$ uniformly, intervals of integer length at least $n^{b}$ have count increase at least $λ / 2$ times their length eventually, and $N (t) / t \to λ$ . Also [F10], a union bound and [F9] make all within-unit Brownian oscillations up to time $C n$ be $O (\sqrt{\log n})$ . These estimates require no independence between evaluation times and the Brownian motion. Consequently the integer coupling in step 3.1 extends to $\sup_{z \leq n} ∣ γ N (z) - μ z - s B_{0} (z) ∣ = o (n^{b})$ , and step 2.1 gives $\max_{k \leq n} ∣ T_{k} - γ N (k) ∣ = o (n^{b})$ .

5.1F9F10step 1.1step 2.1step 4.1constructchoosealgebra

Let $m (t) = \max {k : T_{k} \leq t}$ and $y (t) = \inf {z : N (z) > t / γ}$ . Brownian reflection on dyadic horizons gives $\sup_{z \leq n} ∣ B_{0} (z) ∣ = O (\sqrt{n \log n}) = o (n)$ ; thus step 2.1 implies $T_{k} / k \to μ$ . Together with $N (z) / z \to λ$ , this gives $m (t), y (t) = O (t)$ , absorbing their finite initial values. Put $k = ⌊ y (t) ⌋$ . Monotonicity gives $γ N (k) \leq t < γ N (k + 1)$ , except for an integer endpoint convention which is covered by shifting at most two indices. The unit-count bound in step 4.1 therefore gives $∣ γ N (k) - t ∣ = O (\log n)$ uniformly for $t \leq n$ . On every required range of $O (n)$ indices, $∣ T_{j} - γ N (j) ∣ \leq n^{b}$ eventually by its little-o bound. Choose an integer $J$ with $γ λ J / 2 > 4$ . The lower-increment bound in step 4.1 makes a change of $⌈ J n^{b} ⌉$ indices change $γ N$ by more than $2 n^{b} + O (\log n)$ eventually. The corresponding $T$ values bracket $t$ , using index zero at the lower boundary. Thus $\sup_{t \leq n} ∣ m (t) - y (t) ∣ = O (n^{b})$ . The countable estimates for all deterministic horizon multipliers cover a finite random multiplier as well. This is a direct monotone inverse comparison, not an imported renewal approximation.

6.1F4F7F8F10step 3.1step 4.1step 5.1constructalgebra

Let $E_{i}$ be the interarrival times of $N$ . Their iid exponential law follows from [F7]: for $0 < t_{1} < \dots < t_{k} \leq T$ , sum its ordered density over all total counts $m \geq k$ and integrate the last $m - k$ ordered points above $t_{k}$ . The simplex volume is $(T - t_{k})^{m - k} / (m - k)!$ , so the first- $k$ density is $e^{- λ T} λ^{k} \sum_{j \geq 0} (λ (T - t_{k}))^{j} / j! = λ^{k} e^{- λ t_{k}}$ . Increasing $T$ covers the whole ordered simplex. The spacing change of variables has determinant one and gives $\prod_{i = 1}^{k} λ e^{- λ e_{i}}$ on positive spacings. Hence $E E_{i} = 1 / λ$ , $Var (E_{i}) = 1 / λ^{2}$ and all moments are finite, by elementary exponential integration and integration by parts for integer moments, bounding other positive powers by the next integer power plus one. Apply [F4] to $(E_{i} - 1 / λ, 0)$ with the same auxiliary exponent $r > p$ , and glue its Brownian driver $V$ to the entire already realized arrival sequence by [F8], using a fresh kernel depending only on that sequence. Then $\max_{k \leq n} ∣ \sum_{i = 1}^{k} E_{i} - k / λ - V (k) / λ ∣ = o (n^{1 / r})$ . Since $y (t) = \sum_{i \leq ⌊ t / γ ⌋ + 1} E_{i}$ , set $V_{1} (t) = \sqrt{γ} V (t / γ)$ and use step 4.1's within-unit Brownian bound to obtain $\sup_{t \leq n} ∣ y (t) - t / μ - V_{1} (t) / (λ \sqrt{γ}) ∣ = o (n^{b})$ . The clock sigma-algebra generated by $B_{0}, N, V_{1}$ and their fresh clock randomness remains independent of $B$ : each new kernel depended on $B_{0}$ or on the clock alone and used randomness independent of earlier variables.

7.1F5F10step 2.1step 3.1step 6.1constructalgebra

Apply [F5] to $B$ , $N$ and this entire clock sigma-algebra. It supplies a standard Brownian motion $W^{*}$ independent of that sigma-algebra, hence independent of $V_{1}$ , and $\sup_{z \leq n} ∣ B (z) - λ^{- 1 / 2} W^{*} (N (z)) ∣ = O (\log^{2} (n + 2))$ . Define $G (t) = (v / \sqrt{λ}) W^{*} (t / γ) - (a μ / (λ \sqrt{γ})) V_{1} (t)$ . It is continuous centered Gaussian with independent stationary increments and variance per unit time $v^{2} / (λ γ) + a^{2} μ^{2} / (λ^{2} γ) = (v^{2} + a^{2} s^{2}) / μ = u / μ$ . Thus, when $u > 0$ , $G = κ W$ for a standard Brownian $W$ and $κ = \sqrt{u / μ}$ . When $u = 0$ , positive semidefiniteness forces $a = v = 0$ , so $G = 0$ and adjoin an unused standard $W$ . The case $v = 0$ needs no residual normalization or nonzero Brownian coefficient.

8.1step 2.1step 5.1step 6.1step 7.1constructalgebra

For $y = y (t)$ , the following exact decomposition has eight terms: $F (t) - G (t) = \sum_{i = 1}^{8} Φ_{i} (t)$ , where $Φ_{1} = F (t) - F (T_{m (t)})$ , $Φ_{2} = F (T_{m (t)}) - F (T_{⌊ y ⌋})$ , $Φ_{3} = F (T_{⌊ y ⌋}) - a T_{⌊ y ⌋} + a μ y - v B (y)$ , $Φ_{4} = a (T_{⌊ y ⌋} - γ N (y))$ , $Φ_{5} = - a μ (y - t / μ - V_{1} (t) / (λ \sqrt{γ}))$ , $Φ_{6} = v (B (y) - λ^{- 1 / 2} W^{*} (N (y)))$ , $Φ_{7} = (v / \sqrt{λ}) (W^{*} (N (y)) - W^{*} (t / γ))$ , and $Φ_{8} = a (γ N (y) - t)$ . Adding these expressions cancels every intermediate reward, clock and Gaussian evaluation and leaves exactly $F (t) - G (t)$ . This verifies the signs and deterministic drift, including the bounded last term.

9.1F2F5F6step 2.1step 4.1step 5.1step 6.1step 7.1step 3.2step 8.1algebra

Uniformly for $t \leq n$ , the first term is $o (n^{b})$ by the incomplete-cycle bound in [F2] and step 3.2. The second is $o (n^{b})$ by the inverse comparison in step 5.1 and [F6]. The third is $o (n^{b})$ by step 2.1 at $⌊ y ⌋$ , the bounded fractional-part drift $a μ (y - ⌊ y ⌋)$ , and the within-unit Brownian bound. The fourth is $o (n^{b})$ by the length/Poisson comparison at $⌊ y ⌋$ and the $O (\log n)$ count increase between that point and $y$ . The fifth is $o (n^{b})$ by step 6.1, and the sixth is $O (\log^{2} n) = o (n^{b})$ by step 7.1. The strict inverse of the simple Poisson clock satisfies $N (y (t)) = ⌊ t / γ ⌋ + 1$ , so its distance from $t / γ$ is at most one; step 3.2's fixed-window Brownian bound makes the seventh term $O (\sqrt{\log n}) = o (n^{b})$ , while $∣ Φ_{8} ∣ \leq ∣ a ∣ γ$ . The indices $m (t), y (t)$ are $O (n)$ by step 5.1, so every approximation was evaluated inside its verified range. Therefore $\sup_{t \leq n} ∣ F (t) - κ W (t) ∣ = o (n^{b})$ for $s > 0$ .

10.1F2F3F10step 1.1step 2.1step 3.2step 9.1algebraconstruct

If $s^{2} = 0$ , the iid lengths have variance zero, hence $L_{j} = μ$ for every $j$ simultaneously almost surely. In particular $T_{k} = μ k$ ; positive semidefiniteness of $Γ$ implies $c = 0$ . The joint approximation [F3] gives $F (T_{k}) = \sqrt{u} B (k) + o (n^{b})$ uniformly for $k \leq n$ , with a standard Brownian $B$ when $u > 0$ , and an unused one when $u = 0$ . For $t \leq n$ , $m (t) = ⌊ t / μ ⌋$ , so step 3.2's fixed-window modulus replaces $B (m (t))$ by $B (t / μ)$ at error $o (n^{b})$ . The incomplete-tour bound is again $o (n^{b})$ . Thus $\sup_{t \leq n} ∣ F (t) - \sqrt{u} B (t / μ) ∣ = o (n^{b})$ . The process $W (t) = \sqrt{μ} B (t / μ)$ is standard Brownian by its continuous Gaussian covariance and increments, and the coefficient is the same $κ = \sqrt{u / μ}$ . This proves the tail approximation in every covariance case, including a zero residual or zero full Gaussian covariance, without inverting a singular matrix.

11.1F1F2F10step 3.2step 9.1step 10.1algebra

Restore the finite initial segment of [F2]. Let $A$ be its finite centered reward and $D$ its finite integer length. For $n \geq D$ , the actual sum is $A + F (n - D)$ . Keep the same standard Brownian motion $W$ and use $A + F (n - D) - κ W (n) = A + [F (n - D) - κ W (n - D)] + κ [W (n - D) - W (n)]$ . The fixed-window Brownian estimates in step 3.2 hold on one event for every positive integer window length. On that event choose an integer at least the finite random $D$ ; then $\max_{D \leq k \leq n} ∣ W (k - D) - W (k) ∣ = O (\sqrt{\log n})$ , regardless of any dependence of $D$ and $W$ . Finitely many earlier indices contribute a finite maximum. Steps 9.1 and 10.1 therefore give the complete-path bound $\max_{k \leq n} ∣ \sum_{i = 1}^{k} f (X_{i}) - κ W (k) ∣ = o (n^{b})$ in the factor construction. Neither a moment of the initial law nor an assertion that a randomly shifted Brownian path is standard has been used.

12.1F1F2F11F12step 1.1step 7.1step 10.1step 11.1givenalgebra

Identify $κ$ without making an unsupported second-moment inference from the almost-sure approximation. Use the same factor and fixed post-delay tour law under invariant initial law, as supplied in [F1]–[F2]. Steps 2.1, 3.1, 3.2, 4.1, 5.1, 6.1, 7.1, 8.1, 9.1, 10.1, 11.1 then give a CLT with variance $κ^{2} = u / μ$ : the approximation error divided by $\sqrt{n}$ tends to zero almost surely since $b < 1 / 2$ , and $κ W (n) / \sqrt{n}$ has exactly the centered normal law of variance $κ^{2}$ for each $n$ ; apply [F12]'s converging-together result. Independently, [F11] gives exponential stationary strong mixing of the original observations. Their $Q_{0}$ th moment, with $Q_{0} > 2$ , satisfies [F12]'s moment regime because every positive power of an exponential rate is summable. Hence [F12] gives the stationary CLT with its absolutely convergent covariance-sum variance $σ_{g}^{2}$ , the asymptotic variance in the Statement's convention. The stationary observation-path equality in [F1] makes these two limits refer to the same law. Uniqueness of centered normal characteristic functions, $e^{- t^{2} κ^{2} / 2} = e^{- t^{2} σ_{g}^{2} / 2}$ , gives $κ^{2} = σ_{g}^{2}$ . The assumed positivity therefore gives $κ = σ_{g} > 0$ . This argument uses no uniform-integrability assumption on the approximation error.

13.1A1F1F8F10step 1.1step 11.1step 12.1givenconstructalgebra∎

Finally preserve the actual original chain probability space. The factor construction gives a joint law of its real observation sequence and the rational-time coordinates of the final standard Brownian motion $W$ . Both are standard-Borel real sequence spaces by [F8]. Use [F1]'s lifting assertion, or disintegrate this joint law and apply its conditional kernel at the actual sequence $(f (X_{1}), f (X_{2}), \dots)$ by [F8]; this preserves the entire original probability space and reproduces that observation/Gaussian-coordinate joint law. The approximation event uses only these observations and integer driver coordinates, so it still has probability one. The rational coordinates extend to a continuous Brownian path on a common measurable full-measure event: for each positive integers $M, k$ , some integer $l$ makes all rational pairs in $[0, M]$ at distance below $1 / l$ have coordinate difference below $1 / k$ . This is a Borel condition and has probability one under the original Brownian rational law; completeness of $R$ gives a unique consistent continuous extension on compact intervals. Every fixed real-time evaluation is the limit along a deterministic rational sequence, hence measurable; on the null failure event set the path to zero. Its finite-dimensional laws agree with standard Brownian laws by continuity, so [F10] identifies genuine standard Brownian motion on the extension. With $B = W$ , steps 11.1, 12.1 give error $o (n^{1 / p})$ , and $p > P_{0}$ implies $n^{1 / p} / (n^{1 / P_{0}} \log n) \to 0$ . Choosing the finite eventual bound and starting index in this almost-sure comparison gives precisely the $C (ω), N (ω)$ in the Statement. AC was used through the stated local Gaussian, factor, tour and extension suppliers; the full moment range, arbitrary initial law, positive-variance convention and original observation indices are retained.

Source notes

Flegal–Jones Lemma 3 supplies the stated rate. Its cited older scalar regenerative route does not itself provide the unrestricted moment-range proof. The argument above instead uses the fully local joint adjacent-iid approximation, delayed multistep tour construction and independent Brownian/Poisson composition. The latter's source manuscript statement and roadmap, and the complete almost-sure proof of Theorem 1 in Bashtova–Shashkin's heavy-tail paper motivating the calendar-time decomposition, were retrieved and inspected as recorded in the owner research notes; no external theorem replaces a proof premise here. The zero-variance rate is proved separately in Zero-variance geometric Markov sums have a uniform strong rate; the Statement here retains its explicit positive-variance hypothesis.

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Strong consistency of overlapping batch means

Statement

Let $P$ be a geometrically ergodic Markov chain with invariant probability $π$ , and let the initial distribution be arbitrary. Let $g$ be real Borel and suppose $π (∣ g ∣^{2 + δ + ε}) < \infty$ for some $δ, ε > 0$ . Assume the state-dependent one-step minorization State-dependent one-step minorization: there are measurable $s : X \to [0, 1]$ and a probability measure $Q$ such that $P (x, A) \geq s (x) Q (A)$ for all $x, A$ .

Let $(b_{n})$ be positive integer batch sizes with $1 \leq b_{n} < n$ for all sufficiently large $n$ , $b_{n} \to \infty$ , and $n / b_{n} \to \infty$ . Suppose there is $c \geq 1$ such that $\sum_{n = 1}^{\infty} (b_{n} / n)^{c} < \infty,$ and suppose $\frac{b_{n}}{n} \log n \to 0, \frac{n^{2 / (2 + δ)} (\log n)^{3}}{b_{n}} \to 0,$ there are $n_{0}$ and $c_{1} < \infty$ with $\log n / b_{n} \leq c_{1}$ for every $n \geq n_{0}$ , and $b_{n}^{2} n^{- 2} \log \log n \to 0, b_{n}^{4} n^{- 3} \log \log n \to 0.$

For ${\overset{ˉ}{g}}_{n} = n^{- 1} \sum_{i = 1}^{n} g (X_{i})$ and ${\overset{ˉ}{g}}_{n, j} (b_{n}) = b_{n}^{- 1} \sum_{i = j}^{j + b_{n} - 1} g (X_{i})$ , define ${\hat{σ}}_{O B M, n}^{2} = \frac{n b_{n}}{(n - b_{n}) (n - b_{n} + 1)} \sum_{j = 1}^{n - b_{n} + 1} ({\overset{ˉ}{g}}_{n, j} (b_{n}) - {\overset{ˉ}{g}}_{n})^{2} .$ The strong-consistency conclusion is ${\hat{σ}}_{O B M, n}^{2} ⟶ σ_{g}^{2} almost surely,$ where $σ_{g}^{2}$ is the covariance long-run variance defined by Long-run variance for a stationary Markov chain. The batch sequence satisfies Flegal–Jones overlapping-batch size conditions for the displayed exponent $δ$ ; these conditions are written out above. This corrects the source's incompatible requirement that both integer $b_{n}$ and $n / b_{n}$ be nondecreasing, retaining every numerical batch-size hypothesis.

As in the intended source corollary, if $δ > 2 / 3$ and $(1 + δ / 2)^{- 1} < ν < 3 / 4$ , then $b_{n} = ⌊ n^{ν} ⌋$ satisfies these corrected batch-size hypotheses and has the same strong-consistency conclusion. No monotonicity of $n / ⌊ n^{ν} ⌋$ is asserted.

Facts & Assumptions

Given: The geometrically ergodic chain, its arbitrary initial law, the real measurable observable $g$ , the moment exponents $δ, ε > 0$ , the supplied one-step minorization, and every displayed numerical batch-size hypothesis.

[A1]

AC is assumed through the chain, Brownian, mixing and strong-approximation interfaces. The Axiom of Choice

[F1]

The canonical chain exists for each initial law, and choosing its invariant initial law $π$ makes it strictly stationary. Canonical Markov chain on path space, Invariant initial law makes a Markov chain stationary

[F2]

An invariant kernel with an everywhere finite measurable common geometric total-variation bound admits a geometric envelope integrable under $π$ , on the original arbitrary measurable space. Harris geometric ergodicity admits an integrable total-variation envelope

[F3]

An integrable geometric total-variation envelope gives exponentially decaying stationary strong-mixing coefficients. A measurable observation preserves stationarity and does not increase those coefficients. A total-variation rate bounds strong-mixing coefficients, A measurable observation does not increase stationary mixing coefficients

[F4]

For a centered stationary sequence with a finite $(2 + η)$ th absolute moment, $η > 0$ , summability of $α (n)^{η / (2 + η)}$ gives absolute covariance summability, a finite nonnegative covariance-sum variance, and a CLT with that variance, including zero. Strong-mixing CLTs under bounded, moment, and logarithmic-moment conditions

[F5]

The covariance long-run variance is the stationary covariance sum when absolute summability and the CLT have been established; equality to the CLT variance is a theorem obligation, not a consequence of the definition alone. Long-run variance for a stationary Markov chain

[F6]

Under the stated geometric-ergodicity, minorization and $(2 + δ + ε)$ th moment hypotheses, with positive finite asymptotic standard deviation, the strong invariance principle supplies a Brownian approximation with error $O (n^{1 / (2 + δ)} \log n)$ on an extension of the original chain. Strong invariance principle for geometrically ergodic chains

[F7]

Under the everywhere common geometric total-variation bound, a finite centered $f \in L^{q} (π)$ with zero covariance sum has, for every $2 < p < q$ , the maximal partial-sum rate $o (n^{1 / p})$ under every initial law, without a moment condition on that law. Zero-variance geometric Markov sums have a uniform strong rate

[F8]

A partial-sum approximation by $σ B (n)$ with error $O (n^{α} \log n)$ , $σ \geq 0$ and $0 < α < 1 / 2$ , implies consistency of the exact OBM estimator when $b_{n} / n \to 0$ , $\sum_{n} (b_{n} / n)^{c} < \infty$ , $(b_{n} / n) \log n \to 0$ , and $n^{2 α} (\log n)^{2} / b_{n} \to 0$ . It requires neither stationarity nor batch monotonicity and includes zero variance. Strong Brownian approximation implies overlapping-batch consistency

[F9]

The corrected Flegal–Jones contract retains all numerical rate, summability and endpoint conditions. For $δ > 2 / 3$ and $2 / (2 + δ) < ν < 3 / 4$ , the integer sequence $⌊ n^{ν} ⌋$ satisfies that contract; the definition supplies the floor bounds and each numerical limit. Flegal–Jones overlapping-batch size conditions

[F10]

A standard scalar Brownian motion exists, and the product of its probability space with the original probability space gives an extension preserving the original marginal. Existence and scaling of $d$ -dimensional Brownian motion, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

Proof

1.1A1F1F2F3givenconstructalgebra

Put $q = 2 + δ + ε$ , $μ = π (g)$ and $f = g - μ$ . The moment hypothesis gives $π ∣ g ∣ < \infty$ , since $∣ g ∣ \leq 1 + ∣ g ∣^{q}$ , so $μ$ is finite. The elementary bound $∣ g - μ ∣^{q} \leq 2^{q} (∣ g ∣^{q} + ∣ μ ∣^{q})$ gives $π ∣ f ∣^{q} < \infty$ , and $π f = 0$ . For variance identification consider the canonical chain with initial law $π$ , which is stationary by [F1]. The given geometric bound has an everywhere finite measurable common envelope, so the stronger arbitrary-measurable-space clause of [F2] supplies an integrable such envelope. By [F3], this stationary chain and its observation $f$ have a strong-mixing bound $α_{f} (n) \leq C r^{n}$ with $C < \infty$ and $r \in (0, 1)$ . This construction is only for identifying the stationary covariance parameter; the observed initial law remains arbitrary.

2.1F4F5step 1.1givenalgebra

Take $η = q - 2 = δ + ε > 0$ . The geometric bound in step 1.1 makes $\sum_{n} α_{f} (n)^{η / (2 + η)}$ finite. Apply [F4] to the stationary observed sequence. Its covariance series is absolutely convergent, its sum $v = π (f^{2}) + 2 \sum_{k \geq 1} {Cov}_{π} (f (X_{0}), f (X_{k}))$ is finite and nonnegative, and its CLT has precisely variance $v$ . Thus [F5] identifies $v = σ_{g}^{2}$ , without using the stationary Harris-only formulation of a separate MCMC regime theorem. Write $σ = \sqrt{v} \geq 0$ and $α = 1 / (2 + δ) \in (0, 1 / 2)$ .

3.1F6step 1.1step 2.1given

If $v > 0$ , the original $g$ , the given $δ, ε$ , geometric ergodicity and supplied minorization meet [F6] with the finite positive standard deviation from step 2.1. Hence, for the original arbitrary initial law, on an extension there are a standard Brownian motion $B$ and finite random constants for which $∣ \sum_{i = 1}^{n} f (X_{i}) - σ B (n) ∣ = O (n^{α} \log n)$ almost surely. The same moment exponents and the same covariance variance are used; no stronger moment assumption is introduced.

4.1F7F10step 1.1step 2.1chooseconstructalgebra

If $v = 0$ , choose a real $p$ with $2 + δ < p < q$ , possible because $ε > 0$ . Apply the all-start geometric-envelope clause of [F7] to $f$ ; its moment and zero covariance hypotheses were established in steps 1.1, 2.1. Use the chain indexed from $X_{1}$ as its time-zero chain, whose initial law is still arbitrary. Then $\max_{j \leq n} ∣ \sum_{i = 1}^{j} f (X_{i}) ∣ = o (n^{1 / p})$ almost surely. Since $1 / p < α$ , this implies $∣ \sum_{i = 1}^{n} f (X_{i}) ∣ = O (n^{α} \log n)$ . Adjoin a standard Brownian motion on the product extension supplied by [F10] and set $σ = 0$ . The same approximation as in step 3.1 now holds, with zero Brownian coefficient. No division by $σ$ or implication from a degenerate CLT to an almost-sure rate is used.

5.1F8step 2.1step 3.1step 4.1givenalgebra

In either variance branch the hypotheses of [F8] hold with this $σ, α$ . Indeed $n / b_{n} \to \infty$ gives $b_{n} / n \to 0$ , while the summability and $(b_{n} / n) \log n \to 0$ assumptions are displayed in the Statement. Moreover $n^{2 α} (\log n)^{2} / b_{n} = [n^{2 / (2 + δ)} (\log n)^{3} / b_{n}] / \log n \to 0$ . The other stated numerical endpoint and growth conditions are retained and impose no additional requirement in this reduction. Thus [F8] gives convergence to $σ^{2} = v$ for the exact OBM estimator applied to $Y_{i} = f (X_{i})$ .

6.1F8step 1.1step 2.1step 5.1algebra

Centering the observations by the constant $μ$ does not change this finite estimator: ${\overset{ˉ}{f}}_{n} = {\overset{ˉ}{g}}_{n} - μ$ and ${\overset{ˉ}{f}}_{n, j} (b_{n}) = {\overset{ˉ}{g}}_{n, j} (b_{n}) - μ$ , so their difference equals ${\overset{ˉ}{g}}_{n, j} (b_{n}) - {\overset{ˉ}{g}}_{n}$ . The indexing $j = 0, \dots, n - b_{n}$ in [F8] is exactly $j = 1, \dots, n - b_{n} + 1$ after adding one to the starting index, and its denominator is $(n - b_{n}) (n - b_{n} + 1)$ as here. Therefore its estimator is the displayed ${\hat{σ}}_{O B M, n}^{2}$ , and steps 2.1 and 5.1 prove its almost-sure convergence to $σ_{g}^{2}$ under the original arbitrary initial law.

7.1F7F8F9step 1.1step 2.1step 4.1step 6.1given∎

Finally, the source-corollary range is $2 / (2 + δ) < ν < 3 / 4$ . By [F9], $b_{n} = ⌊ n^{ν} ⌋$ satisfies every corrected numerical batch hypothesis, so steps 1.1, 2.1, 3.1, 4.1, 5.1, 6.1 apply to it as well. This verifies the intended floor-power conclusion without asserting the incompatible ratio monotonicity. The proof includes $v = 0$ , finite early inadmissible batch indices, and arbitrary initial laws without additional moment hypotheses, as allowed by [F7]–[F8].

5 · Examples, counterexamples and false statements

None yet.