TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

Depends on

Used by

Strong consistency of overlapping batch means Theorem
Strong invariance principle for geometrically ergodic chains Theorem
Sufficient MCMC CLT regimes Theorem

Dependency tree · two levels

62 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Galin L. Jones, On the Markov Chain Central Limit Theorem, Theorems 5 and 6 (standard reference, not scraped)
P. Doukhan, P. Massart and E. Rio, The functional central limit theorem for strongly mixing processes, Lemma 1, Theorem 1 and applications (standard reference, not scraped)