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Moving block bootstrap CLT for dependent Hilbert-valued data

Statement

Assume the Axiom of Choice. Let $H = L^{2} ([0, 1])$ , and let $(X_{t})_{t \in Z}$ be a centered causal Bernoulli shift that is $L^{2}$ - $m$ -approximable under L2-m-approximable causal Bernoulli shifts. Write

$δ_{m} = (E ∥ X_{0} - X_{0, m} ∥_{H}^{2})^{1 / 2} .$

By Independent-tail replacement errors are decreasing and summability implies the rate condition, the summability in the stated process definition implies $m δ_{m} \to 0$ . For each $n$ , let $b_{n}$ be an integer with $b_{n} \to \infty$ and $b_{n} / \sqrt{n} \to 0$ . For all sufficiently large $n$ , these conditions give $1 \leq b_{n} \leq n$ ; apply Moving block bootstrap to $X_{1}, \dots, X_{n}$ on this admissible tail. All limits below concern that tail. Let $Γ$ be the long-run covariance operator from Hilbert-valued CLT for L2-m-approximable Bernoulli shifts, and let ${\overset{ˉ}{X}}_{n}^{*} = n^{- 1} \sum_{t = 1}^{n} X_{t}^{*}$ . Then

$L^{*} ⁣ (\sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - E^{*} {\overset{ˉ}{X}}_{n}^{*})) \Rightarrow L (G) in probability,$

and

${∥ n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*}) - Γ ∥}_{H S} ⟶ 0 in probability,$

where $G$ is the centered $H$ -valued Gaussian with covariance $Γ$ .

Facts & Assumptions

Given: AC, the centered causal shift with summable replacement errors, and $b = b_{n} \to \infty$ , $b / \sqrt{n} \to 0$ .

[F1]

L2-m-approximable causal Bernoulli shifts supplies the centered measurable causal representation, second moment, replacement extension and summable errors.

[F2]

Hilbert-valued CLT for L2-m-approximable Bernoulli shifts supplies the sampling-law limit $n^{- 1 / 2} \sum_{t = 1}^{n} X_{t} \Rightarrow G$ and its long-run covariance $Γ$ . This is a sample-sum CLT, not a time-indexed FCLT.

[F3]

Moving block bootstrap concatenates independent uniformly chosen overlapping blocks and retains exactly $n$ observations. Exact conditional covariance of a moving block bootstrap mean gives its exact full-plus-partial-block covariance formula.

[F4]

Weighted overlapping-block moment and covariance limits proves uniform integrability and empirical block covariance consistency for bounded normalized deterministic weights with fixed-lag autocorrelations tending to one. Conditional CLT for independent resampled weighted blocks proves the conditional CLT, tightness and trace-norm covariance limit for exactly this resampling, including the last partial block.

[F5]

Under the independent-tail construction in [F1], summability of the replacement errors implies $m δ_{m} \to 0$ (Independent-tail replacement errors are decreasing and summability implies the rate condition).

[A1]

The Axiom of Choice is assumed. Its uses are those of the Hilbert sampling and conditional-block suppliers, including the complete orthonormal basis in the latter; no further choice principle is added here.

Proof

Given: The assumptions and suppliers above.

1.1givenF1F4F5A1algebra

Take $c_{b, ℓ} = 1$ for $1 \leq ℓ \leq b$ . These weights are bounded by one, have $b^{- 1} \sum_{ℓ} c_{b, ℓ}^{2} = 1$ , and for each fixed $h \geq 0$ have $b^{- 1} \sum_{ℓ = 1}^{b - h} c_{b, ℓ} c_{b, ℓ + h} = (1 - h / b)_{+} \to 1$ . By [F5], the process satisfies the rate needed in [F4], and the given block regime supplies its remaining asymptotic condition.

2.1F3F4step 1.1algebra

Write $n = k b + r$ , $0 \leq r < b$ . Subtracting the conditional mean from the sum of the $k$ complete sampled blocks and the sampled length- $r$ prefix gives exactly the variable $Z_{n}^{*}$ in the conditional-block supplier. The candidate starts for both block sizes are the same $N = n - b + 1$ starts, as required by [F3]. Hence [F4] gives the asserted conditional law and trace-norm covariance convergence. The Hilbert-Schmidt conclusion follows from $∥ T ∥_{H S} \leq ∥ T ∥_{1}$ ; the covariance in question is exactly $n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*})$ by [F3]. The remainder has expected conditional squared norm $O (b / n)$ , so no divisibility condition is used.

3.1F2F4step 2.1∎

Fix the sampling-law Gaussian $G$ supplied by [F2]. The conditional-block result [F4] applies with this same centered Gaussian, whose covariance is $Γ$ . This proves both stated limits.

The paper's Assumption 1 states $m δ_{m} \to 0$ ; the local independent-tail replacement lemma derives this rate from summability under the stated replacement convention, so the local block argument adds no process restriction.

Depends on

Used by

IID functional data in the block bootstrap regime Example

Dependency tree · two levels

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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), Theorem 2.1 and Appendix (standard reference, not scraped)
A. DasGupta, Asymptotic Theory of Statistics and Probability, Section 29.18, Block Bootstrap Methods, pp. 478-479 (standard reference, not scraped)