PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Bounded profile functional conditional CLT

Statement

Let $(Ω, F, P)$ be the common probability space for the innovations and their independent replacement family (take a countable independent product extension if needed), and write $E$ for expectation under $P$ .

Assume the Axiom of Choice. Let $H = L^{2} ([0, 1], B, λ; R)$ , where $B$ is generated by intervals with rational endpoints and $λ$ is Lebesgue measure restricted to $B$ . Let $(S, S, ν)$ be a probability space, let $(δ_{t})_{t \in Z}$ be independent identically distributed measurable $S$ -valued random elements with law $ν$ , and let $f : S^{N_{0}} \to H$ be measurable from the product sigma-algebra $S^{\otimes N_{0}}$ to the Borel sigma-algebra of $H$ . Set $X_{t} = f (δ_{t}, δ_{t - 1}, \dots)$ and assume its Bochner expectation is zero and $E ∥ X_{0} ∥_{H}^{2} < \infty$ . For each $t \in Z$ , $m \geq 1$ , and $k \leq t - m$ , let $δ_{t, m, k}^{'}$ have law $ν$ , with these copies mutually independent and independent of the original innovations, and set $X_{t, m} = f (δ_{t}, \dots, δ_{t - m + 1}, δ_{t, m, t - m}^{'}, δ_{t, m, t - m - 1}^{'}, \dots)$ . Assume $\sum_{m = 1}^{\infty} δ_{m} < \infty$ , where $δ_{m} = (E ∥ X_{0} - X_{0, m} ∥_{H}^{2})^{1 / 2}$ . For each $n \geq 1$ , let the integer block length $b_{n}$ satisfy $b_{n} \to \infty$ and $b_{n} / \sqrt{n} \to 0$ ; for all sufficiently large $n$ , apply moving-block resampling with block length $b_{n}$ to $X_{1}, \dots, X_{n}$ , truncating the last block to exactly $n$ elements. Write ${\overset{ˉ}{X}}_{n}^{*} = n^{- 1} \sum_{j = 1}^{n} X_{j}^{*}$ , and let $E^{*}$ denote expectation over these draws with the observations fixed. For a finite-support $H$ -valued resampling random element $Z$ , define its conditional covariance operator by

${Cov}^{*} (Z) : = E^{*} [(Z - E^{*} Z) \otimes (Z - E^{*} Z)] \in B (H), (u \otimes v) h : = ⟨ h, v ⟩_{H} u (h, u, v \in H) .$

For a finite-support real-valued resampling random variable $V$ , define ${Var}^{*} (V) : = E^{*} [(V - E^{*} V)^{2}]$ . Each convergence-in-probability assertion below is with respect to $P$ .

Write $L^{*} (Z)$ for the law of $Z$ under the same conditional resampling distribution. Let $G$ be the centered $H$ -valued Gaussian limit with covariance operator $Γ$ supplied by the Hilbert-valued sample-mean CLT for this process. For any bounded real-linear $L : H \to R$ , set $∥ L ∥ = \sup_{∥ u ∥_{H} \leq 1} ∣ L (u) ∣$ , let $h_{L} \in H$ be the unique vector satisfying $L (u) = ⟨ u, h_{L} ⟩_{H}$ for every $u \in H$ , and define $L^{*} r = r h_{L}$ . Identify the scalar operator $L Γ L^{*} : R \to R$ with its value at $1$ . Then $L^{*} ⁣ (\sqrt{n} L ({\overset{ˉ}{X}}_{n}^{*} - E^{*} {\overset{ˉ}{X}}_{n}^{*})) \Rightarrow L (L G) in probability,$ and, with $C_{n} = n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*})$ , $∣ {Var}^{*} ⁣ (\sqrt{n} L ({\overset{ˉ}{X}}_{n}^{*} - E^{*} {\overset{ˉ}{X}}_{n}^{*})) - L Γ L^{*} ∣ \leq ∥ L ∥^{2} ∥ C_{n} - Γ ∥_{H S} ⟶ 0 in probability .$ Here $L Γ L^{*}$ is the variance of the limiting scalar Gaussian. This conclusion concerns bounded linear profile observations under the stated Bernoulli-shift and block-length hypotheses; it makes no claim for nonlinear statistics or for finite-replicate quantile accuracy.

Statement references

Profile-to-Hilbert data, resampling law, and conditional center

Conditional Hilbert-valued moving-block bootstrap CLT

Hilbert-valued sample-mean CLT and long-run covariance

Riesz representative of a bounded functional

Facts & Assumptions

Given: The statement assumptions, including the Axiom of Choice and the exact block-length regime.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; it is the stated assumption of the moving-block theorem below.

[F1]

Profile-to-Hilbert bootstrap interface: $H = L^{2} ([0, 1], B, λ; R)$ is the real Hilbert space of almost-everywhere classes; the measurable causal process has Bochner mean $E_{B} X_{0} : = \int_{Ω} X_{0} d P = 0$ and finite second moment; its replacement family lies on an independent product extension whose stated setup assumes Countable Choice and Dependent Choice, with $δ_{m} = (E ∥ X_{0} - X_{0, m} ∥_{H}^{2})^{1 / 2}$ and $\sum_{m \geq 1} δ_{m} < \infty$ ; MBB uses independent uniform overlapping starts, preserves within-block order, truncates to length $n$ , and is centered by its actual conditional expectation.

[F2]

Moving block bootstrap CLT for dependent Hilbert-valued data: under AC, centered causal $L^{2}$ - $m$ -approximable $H$ -valued data and $b_{n} \to \infty, b_{n} / \sqrt{n} \to 0$ , the conditional law of $\sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - E^{*} {\overset{ˉ}{X}}_{n}^{*})$ converges weakly in probability to $L (G)$ , and $∥ n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*}) - Γ ∥_{H S} \to 0$ in probability.

[F3]

Hilbert-valued CLT for L2-m-approximable Bernoulli shifts: the sampling-law limit $G$ is centered Gaussian in $H$ , with covariance $Γ$ ; the result is a sample-sum CLT, not a time-indexed functional CLT.

[F4]

Conditional weak convergence in probability: convergence in conditional weak topology in probability means convergence in probability in a metric that metrizes weak convergence. Weak convergence of borel probability measures characterizes weak convergence by bounded continuous test functions.

[F5]

Riesz representation for Hilbert spaces: under Countable Choice, every bounded real-linear functional on a real Hilbert space has a unique representing vector $h_{L}$ , with $∥ h_{L} ∥_{H} = ∥ L ∥$ ; thus $L^{*} r = r h_{L}$ and $∥ L^{*} ∥ = ∥ L ∥$ .

[F6]

L2-m-approximable causal Bernoulli shifts supplies separability of $H$ under Countable Choice, and A Hilbert space with a dense sequence has a finite or countable orthonormal basis turns a fixed countable dense sequence into a finite or countable orthonormal basis.

[F7]

Hilbert–Schmidt operator and Hilbert–Schmidt norm defines the Hilbert-Schmidt norm relative to a supplied Hilbert basis; The Hilbert–Schmidt norm is basis independent makes its unqualified value independent of that basis; and Hilbert Schmidt operators form a two sided ideal gives $∥ A T B ∥_{H S} \leq ∥ A ∥ ∥ T ∥_{H S} ∥ B ∥$ for bounded $A, B$ . On $R$ , the Hilbert-Schmidt norm of a scalar operator is its absolute value.

[F8]

AC implies DC implies countable choice: in ZF, the Axiom of Choice implies Dependent Choice and then Countable Choice, with Dependent Choice formulated using a prescribed initial point. Together with [A1], this supplies the Countable Choice and Dependent Choice assumptions in [F1], and the Countable Choice assumptions in [F5] and [F6].

[F9]

Exact conditional covariance of a moving block bootstrap mean uses the conditional covariance operator and rank-one convention stated above and proves the exact finite-sample MBB covariance identity for fixed Hilbert-valued observations.

Proof

Given: The assumptions and supplier facts above.

1.1A1F1F2F4F6F8given

By [A1] and [F8], AC yields Countable Choice and Dependent Choice. By [F1] and [F6], $H$ is separable complete. Put $μ_{n}^{*} = L^{*} (\sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - E^{*} {\overset{ˉ}{X}}_{n}^{*}))$ and $μ = L (G)$ , and let $d_{H}$ be the metric on Borel probability laws on $H$ used in [F2] under [F4]. Thus $d_{H}$ metrizes weak convergence and $d_{H} (μ_{n}^{*}, μ) \to 0$ in probability. For a continuous map $T : E \to F$ between metric spaces and a Borel law $η$ on $E$ , write $T_{#} η (B) = η (T^{- 1} (B))$ for Borel $B \subseteq F$ . Composing bounded continuous tests with $T$ in [F4] shows that $η_{j} \Rightarrow η$ implies $T_{#} η_{j} \Rightarrow T_{#} η$ .

2.1F1F4step 1.1constructalgebra

To supply the scalar-law metric required by [F4], let $e = [ξ \mapsto 1] \in H$ , $J : R \to H$ , $J (r) = r e$ , and $P : H \to R$ , $P (u) = ⟨ u, e ⟩_{H}$ . Here $∥ e ∥_{H} = 1$ , $J$ is an isometry, and $P$ is continuous: expanding $∥ u - P (u) e ∥_{H}^{2} \geq 0$ gives $∣ P (u) ∣ \leq ∥ u ∥_{H}$ . Also $P J$ is the identity. For Borel probability laws $α, β$ on $R$ , define $d_{R} (α, β) = d_{H} (J_{#} α, J_{#} β)$ . This is a metric: symmetry and the triangle inequality come from $d_{H}$ , and $J_{#} α = J_{#} β$ implies $α = P_{#} J_{#} α = P_{#} J_{#} β = β$ . By step 1.1 applied to $J$ and $P$ , $α_{j} \Rightarrow α$ if and only if $J_{#} α_{j} \Rightarrow J_{#} α$ ; hence $d_{R}$ metrizes scalar weak convergence.

3.1F1F2F4step 1.1step 2.1algebra

Step 1.1 applied to $L$ shows that $L_{#}$ is sequentially continuous for $d_{H}$ and $d_{R}$ , hence continuous: otherwise a sequence within $1 / j$ of a point would have images outside one fixed positive-radius ball. Thus for every $ε > 0$ there is $δ > 0$ such that $d_{H} (η, μ) < δ$ implies $d_{R} (L_{#} η, L_{#} μ) < ε$ , and $P (d_{R} (L_{#} μ_{n}^{*}, L_{#} μ) \geq ε) \leq P (d_{H} (μ_{n}^{*}, μ) \geq δ) \to 0$ . The scalar metric distance is measurable: for fixed $n$ the finitely many block-start tuples have fixed weights and atoms continuous in the data $(X_{1}, \dots, X_{n}) \in H^{n}$ ; their law is weakly continuous by bounded continuous tests, hence continuous for $d_{R}$ . Conditional expectation is the corresponding finite weighted sum, so it commutes with $L$ . Therefore $L_{#} μ_{n}^{*}$ is the displayed conditional scalar law and $L_{#} μ = L (L G)$ , proving the first conclusion in precisely the sense of [F4].

4.1A1F1F2F3F5F6F7F8F9step 1.1algebra∎

By [A1] and [F8], the assumed AC supplies Countable Choice and Dependent Choice. Then [F5] gives the Riesz representative, while [F1] and [F6] give separability of $H$ and a countable orthonormal basis; use ${1}$ as the basis of $R$ . Put $T_{n} = C_{n} - Γ$ and $Z_{n}^{*} = \sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - E^{*} {\overset{ˉ}{X}}_{n}^{*})$ . By the covariance definition in the Statement, ${Cov}^{*} (Z_{n}^{*}) = C_{n}$ , since centering and scaling by $\sqrt{n}$ multiplies covariance by $n$ . The Riesz identity and the same conditional covariance convention as [F9] give ${Var}^{*} (L (Z_{n}^{*})) = E^{*} ⟨ Z_{n}^{*}, h_{L} ⟩_{H}^{2} = ⟨ C_{n} h_{L}, h_{L} ⟩_{H} = (L C_{n} L^{*}) (1)$ . Apply [F7] to $A = L$ , $T = T_{n}$ , and $B = L^{*}$ : $∣ L T_{n} L^{*} (1) ∣ = ∥ L T_{n} L^{*} ∥_{H S} \leq ∥ L ∥ ∥ T_{n} ∥_{H S} ∥ L^{*} ∥ = ∥ L ∥^{2} ∥ T_{n} ∥_{H S}$ . By [F2] this bound tends to zero in probability. The covariance of $L G$ is $L Γ L^{*}$ by [F3], proving the variance conclusion, including the zero-variance case.

Bounded profile functional conditional CLT

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources