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

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Conditional CLT for independent resampled weighted blocks

Statement

Assume AC and all the process and weight hypotheses of Weighted overlapping-block moment and covariance limits, including squared normalization and convergence of each fixed-lag weight autocorrelation to one. Let $b = b_{n} \to \infty$ with $b / \sqrt{n} \to 0$ , $N = n - b + 1$ , $k = ⌊ n / b ⌋$ , $r = n - k b$ , and

$B_{j, q} = \sum_{ℓ = 1}^{q} c_{b, ℓ} X_{j + ℓ - 1}, {\overset{ˉ}{B}}_{q} = N^{- 1} \sum_{j = 1}^{N} B_{j, q}, C_{q} = N^{- 1} \sum_{j = 1}^{N} (B_{j, q} - {\overset{ˉ}{B}}_{q}) \otimes (B_{j, q} - {\overset{ˉ}{B}}_{q}),$

with $B_{j, 0} = {\overset{ˉ}{B}}_{0} = C_{0} = 0$ . Conditional on the observations choose independent uniform starts $J_{1}, \dots, J_{k + 1}$ in ${1, \dots, N}$ and set

$Z_{n}^{*} = n^{- 1 / 2} {\sum_{s = 1}^{k} (B_{J_{s}, b} - {\overset{ˉ}{B}}_{b}) + (B_{J_{k + 1}, r} - {\overset{ˉ}{B}}_{r})} .$

Then, for the centered Hilbert Gaussian $G$ of covariance $Γ$ ,

$L^{*} (Z_{n}^{*}) \Rightarrow L (G) in probability, {∥ {Cov}^{*} (Z_{n}^{*}) - Γ ∥}_{1} ⟶ 0 in probability .$

Trace-norm convergence implies convergence in Hilbert-Schmidt norm as well. The conditional laws are tight in probability: for every $ε, η > 0$ there is a deterministic compact $K \subset H$ such that ${lim sup}_{n} P {P^{*} (Z_{n}^{*} \notin K) > ε} < η$ . The complete blocks satisfy conditional Lindeberg, and the final partial block has $E E^{*} ∥ n^{- 1 / 2} (B_{J_{k + 1}, r} - {\overset{ˉ}{B}}_{r}) ∥^{2} = O (b / n)$ . Replacing every $X_{t}$ by $X_{t} - M_{n}$ for any common data-dependent $M_{n} \in H$ leaves $Z_{n}^{*}$ and all its conditional covariances unchanged.

Facts & Assumptions

Given: The assumptions and the conditional uniform-start randomization of the Statement. Write $∥ \cdot ∥_{1}$ for trace norm.

[F1]

Weighted overlapping-block moment and covariance limits supplies uniform integrability of $∥ b^{- 1 / 2} B_{1, b} ∥^{2}$ , $C_{b} / b \to Γ$ in trace norm in probability, $E ∥ {\overset{ˉ}{B}}_{b} / \sqrt{b} ∥^{2} = O (b / N)$ , and $E ∥ B_{1, q} ∥^{2} \leq D q$ uniformly for $q \leq b$ . For any complete orthonormal basis and its projections $P_{d}$ , putting $R_{d} = I - P_{d}$ , it also supplies $E ∥ R_{d} B_{1, q} ∥^{2} \leq q D_{d}$ with $D_{d} \to 0$ .

[F2]

Hilbert-valued CLT for L2-m-approximable Bernoulli shifts supplies $G$ with positive trace-class covariance $Γ$ . Under AC choose a complete eigenbasis for $Γ$ by Orthonormal eigenbasis for a compact self adjoint operator. Thus $E ∥ R_{d} G ∥^{2} = tr (R_{d} Γ R_{d}) \to 0$ . AC is used for these inherited Hilbert-space constructions, not for an additional uncountable randomization.

[F3]

Second-order characteristic-function expansion gives $∣ e^{i u} - 1 - i u + u^{2} / 2 ∣ \leq \min (∣ u ∣^{3} / 3, 4 u^{2})$ . A centered normal of variance $v \geq 0$ has characteristic function $e^{- t^{2} v / 2}$ by Characteristic function of a normal law, and the characteristic function of a finite independent sum is the product of the individual characteristic functions (Characteristic functions under affine maps and independent sums). The characteristic-function criterion and Cramer--Wold apply also to degenerate limits (Characteristic function criterion for weak convergence, Cramer wold device).

[F4]

A scalar sequence converges in probability exactly when every subsequence has an almost-surely convergent further subsequence (Subsequence characterization of convergence in probability, An almost-surely convergent subsequence from convergence in probability). Weak convergence on a metric space is equivalent to convergence against all bounded uniformly continuous tests by Portmanteau theorem. Under the assumed AC, Levy prokhorov metric metrizes weak convergence supplies the Levy-Prokhorov metric generating weak convergence on the separable Hilbert space $H$ . Use this metric in Conditional weak convergence in probability. For each $n$ the conditional law is a finite mixture with deterministic probabilities and Borel data-dependent atoms: finite independent starts have only finitely many configurations. The map from its atom tuple to this law is continuous for weak convergence (test each bounded continuous function), hence its metric distance to the fixed law is a Borel data function.

[F5]

Markov's inequality is Markov's inequality for random variables. The trace-class ideal estimate is Trace class is a two sided Banach operator ideal, and Nuclear series characterizes trace norm identifies the trace norm with the infimum of nuclear sums. The Hilbert-Schmidt norm is the basis square-sum in Hilbert–Schmidt operator and Hilbert–Schmidt norm; The finite Bessel inequality and best approximation by a finite orthonormal family bounds finite sums of squared coordinates. These give the operator bounds proved in step 1.1.

[F6]

Hilbert-space Cauchy–Schwarz gives $∣ ⟨ a, x ⟩_{H} ∣ \leq ∥ a ∥_{H} ∥ x ∥_{H}$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A1]

The Axiom of Choice is assumed as in [F2].

Proof

Given: The stated process, weights, data and independent conditional block starts.

1.1givenF1F5algebra

First, for a nuclear representation $T = \sum_{j} ⟨ \cdot, u_{j} ⟩ v_{j}$ and an orthogonal projection $R$ , its absolutely convergent trace has $∣ tr (R T R) ∣ \leq \sum_{j} ∥ u_{j} ∥ ∥ v_{j} ∥$ . For every finite orthonormal set $E$ , Minkowski and Bessel give $(\sum_{e \in E} ∥ T e ∥^{2})^{1 / 2} \leq \sum_{j} ∥ v_{j} ∥ (\sum_{e \in E} ∣ ⟨ e, u_{j} ⟩ ∣^{2})^{1 / 2} \leq \sum_{j} ∥ u_{j} ∥ ∥ v_{j} ∥$ . Infinite nuclear sums follow from their summable tails. Taking the nuclear infimum proves $∣ tr (R T R) ∣ \leq ∥ T ∥_{1}$ and $∥ T ∥_{H S} \leq ∥ T ∥_{1}$ by [F5]. Subtraction of $M_{n}$ replaces $B_{j, q}$ by $B_{j, q} - s_{q} M_{n}$ , where $s_{q} = \sum_{ℓ = 1}^{q} c_{b, ℓ}$ does not depend on $j$ . It changes ${\overset{ˉ}{B}}_{q}$ by exactly the same amount. Thus every centered block is unchanged. Independence of the starts gives ${Cov}^{*} (Z_{n}^{*}) = (k C_{b} + C_{r}) / n$ . Positivity and the rank-one trace formula give $E tr C_{r} \leq E ∥ B_{1, r} ∥^{2} \leq D r$ , so $E ∥ C_{r} / n ∥_{1} \leq D b / n \to 0$ . Since $k b / n \to 1$ , [F1] now gives the asserted trace-norm covariance limit. The conditional squared norm of the centered remainder has expectation $tr (C_{r} / n)$ and therefore its unconditional expectation tends to zero at rate $O (b / n)$ .

2.1F1F5step 1.1algebra

Set $U_{j} = B_{j, b} / \sqrt{b}$ , $\overset{ˉ}{U} = {\overset{ˉ}{B}}_{b} / \sqrt{b}$ , and $V_{j} = U_{j} - \overset{ˉ}{U}$ . For any vectors $x, y$ and $a > 0$ , splitting according as the larger summand exceeds $a / 2$ gives $∥ x + y ∥^{2} 1_{{∥ x + y ∥ > a}} \leq 4 ∥ x ∥^{2} 1_{{∥ x ∥ > a / 2}} + 4 ∥ y ∥^{2} 1_{{∥ y ∥ > a / 2}}$ . Consequently, for each $ε > 0$ , the conditional Lindeberg sum of the $k$ complete blocks is at most $L_{n} (ε) = \frac{k b}{n N} \sum_{j = 1}^{N} ∥ V_{j} ∥^{2} 1_{{∥ V_{j} ∥ > ε \sqrt{n / b}}}, E L_{n} (ε) \leq 4 E ⁣ [∥ U_{1} ∥^{2} 1_{{∥ U_{1} ∥ > (ε / 2) \sqrt{n / b}}}] + 4 E ∥ \overset{ˉ}{U} ∥^{2} ⟶ 0.$ Here $n / b \to \infty$ , stationarity makes the expected empirical uncentered tail equal to that of $U_{1}$ , and [F1] supplies uniform integrability and $E ∥ \overset{ˉ}{U} ∥^{2} \to 0$ . Hence $L_{n} (ε) \to 0$ in probability by [F5].

3.1F2F3F4F6step 1.1step 2.1choosealgebra

Take any subsequence. Apply [F4] diagonally to the scalar trace-norm covariance error, $tr (C_{r} / n)$ , and $L_{n} (1 / j)$ for positive integers $j$ . There is a further subsequence on which all tend to zero almost surely. Fix a data outcome in this probability-one event. For any $a \in H$ , let $ξ_{n} = ⟨ a, B_{J_{1}, b} - {\overset{ˉ}{B}}_{b} ⟩ / \sqrt{n}$ and $v_{n} = k E^{*} ξ_{n}^{2}$ . Step 1.1 and remainder negligibility give $v_{n} \to ⟨ Γ a, a ⟩$ , and the norm Lindeberg bounds imply $k E^{*} [ξ_{n}^{2} 1_{{∣ ξ_{n} ∣ > ρ}}] \to 0$ for every $ρ > 0$ . The Taylor bound in [F3], split at $ρ$ , gives $k ∣ E^{*} e^{i ξ_{n}} - 1 + \frac{1}{2} E^{*} ξ_{n}^{2} ∣ \leq (ρ / 3) v_{n} + 4 k E^{*} [ξ_{n}^{2} 1_{{∣ ξ_{n} ∣ > ρ}}] .$ Let $n \to \infty$ and then $ρ ↓ 0$ . Thus $k (E^{*} e^{i ξ_{n}} - 1) \to - ⟨ Γ a, a ⟩ / 2$ and $k ∣ E^{*} e^{i ξ_{n}} - 1 ∣^{2} \to 0$ , since $k \to \infty$ . The convergent power-series expansion $\log (1 + z) = z + O (∣ z ∣^{2})$ near zero proves $(E^{*} e^{i ξ_{n}})^{k} \to \exp (- ⟨ Γ a, a ⟩ / 2)$ . The remainder's characteristic function tends to one because its conditional mean is zero and [F3], [F6] give $∣ E^{*} e^{i ⟨ a, R_{n} ⟩} - 1 ∣ \leq E^{*} ⟨ a, R_{n} ⟩^{2} \leq ∥ a ∥_{H}^{2} tr (C_{r} / n) \to 0$ , where $R_{n} = n^{- 1 / 2} (B_{J_{k + 1}, r} - {\overset{ˉ}{B}}_{r})$ . Replacing $a$ by $t a$ covers every real frequency. Therefore [F3] gives weak convergence of every finite-dimensional projection of the conditional law to that of $G$ , on this fixed outcome; no positive lower bound on any variance was used.

4.1F2F4F5step 1.1step 3.1algebra

On the same outcome let $T_{n} = {Cov}^{*} (Z_{n}^{*})$ . For every fixed $d$ , conditional centering and [F5] give $E^{*} ∥ R_{d} Z_{n}^{*} ∥^{2} = tr (R_{d} T_{n} R_{d}) \to tr (R_{d} Γ R_{d})$ . If $ϕ$ is bounded uniformly continuous, put $M = \sup ∣ ϕ ∣$ and $ω_{ϕ} (h) = \sup_{∥ x - y ∥ \leq h} ∣ ϕ (x) - ϕ (y) ∣$ . Markov gives $∣ E^{*} ϕ (Z_{n}^{*}) - E^{*} ϕ (P_{d} Z_{n}^{*}) ∣ \leq ω_{ϕ} (h) + 2 M h^{- 2} tr (R_{d} T_{n} R_{d}) .$ The analogous bound for $G$ uses $tr (R_{d} Γ R_{d})$ . Step 3.1 gives convergence of the projected expectations. Take the limit superior in $n$ , then $d \to \infty$ , and finally $h ↓ 0$ . All such test expectations converge, so [F4] gives weak convergence of the full conditional laws on this outcome. Thus their distance to the law of $G$ in the metric of [F4] tends to zero almost surely on the extracted subsequence. The subsequence characterization proves convergence in probability along the original sequence.

5.1A1F1F5step 1.1step 4.1choosealgebra∎

To verify tightness explicitly, empirical centering decreases each projected second moment. By [F1] and independence, $E E^{*} ∥ R_{d} Z_{n}^{*} ∥^{2} \leq \frac{k b + r}{n} D_{d} = D_{d} .$ The same bound without projection is $D$ . Enlarge constants to cover any finitely many initial admissible block lengths. Given $ε, η > 0$ , choose $R$ and increasing integers $d_{j}$ so that $D / R^{2} < ε η / 4$ and $j^{2} D_{d_{j}} < ε η 2^{- j - 2}$ . The closed set $K = {x \in H : ∥ x ∥ \leq R, ∥ R_{d_{j}} x ∥ \leq 1 / j for every j \geq 1}$ is compact: for any radius choose $j$ with $1 / j$ smaller than half the radius, and a finite net for the bounded ball in $P_{d_{j}} H$ ; this gives a finite net for $K$ , and closedness in complete $H$ gives compactness. Markov and the union bound yield $E P^{*} (Z_{n}^{*} \notin K) < ε η / 2$ , and then $P {P^{*} (Z_{n}^{*} \notin K) > ε} < η / 2$ . This proves the stated conditional tightness, and steps 1.1-4.1 establish all the other assertions.

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Sources

D. Pilavakis, E. Paparoditis, and T. Sapatinas, Moving block and tapered block bootstrap for functional time series with an application to the K-sample mean problem, Bernoulli 25 (2019), proof of Theorem 2.1; weighted-block and arbitrary-length extension proved locally (standard reference, not scraped)