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Tapered block bootstrap CLT for dependent Hilbert-valued data

Statement

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

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

By Independent-tail replacement errors are decreasing and summability implies the rate condition, the summability in the stated process definition implies $m δ_{m} \to 0$ . Let the integers $b_{n} \to \infty$ with $b_{n} / \sqrt{n} \to 0$ . Let the taper satisfy the support, positivity-near- $1 / 2$ , symmetry and monotonicity conditions in Tapered block bootstrap. For all sufficiently large $n$ , $1 \leq b_{n} \leq n$ and $W_{b_{n}} > 0$ , the latter by Asymptotic squared-norm normalization of a tapered window. Apply that algorithm to $Y_{1}, \dots, Y_{n}$ only on this admissible tail; all limits below concern that tail. Write ${\overset{ˉ}{Y}}_{n}^{*} = n^{- 1} \sum_{t = 1}^{n} Y_{t}^{*}$ for the resulting bootstrap mean. Let $Γ$ be the long-run covariance operator from Hilbert-valued CLT for L2-m-approximable Bernoulli shifts. Then

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

and

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

where $G$ is centered Gaussian with covariance $Γ$ .

Facts & Assumptions

Given: AC; the centered causal shift with summable replacement errors; the stated block regime; and the admissible taper.

[F1]

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

[F2]

Hilbert-valued CLT for L2-m-approximable Bernoulli shifts supplies the sampling-law limit and centered Gaussian $G$ of covariance $Γ$ ; it makes no time-indexed FCLT assertion.

[F3]

Tapered block bootstrap uses midpoint weights $c_{b, ℓ} = a_{b} w ((ℓ - 1 / 2) / b)$ on sample-centered observations, independent uniform starts and truncation to exactly $n$ observations. Exact conditional covariance of a tapered block bootstrap mean gives the exact conditional full-plus-partial covariance.

[F4]

Asymptotic squared-norm normalization of a tapered window gives $W_{b}^{2} / b \to I_{w} > 0$ , so $a_{b} = \sqrt{b} / W_{b}$ is defined and bounded for all sufficiently large $b$ .

[F5]

Weighted overlapping-block moment and covariance limits proves uniform integrability and empirical trace-norm covariance consistency for bounded normalized weights with fixed-lag autocorrelations tending to one. Conditional CLT for independent resampled weighted blocks proves the corresponding conditional law, tightness, trace-norm covariance limit and partial-block bound; subtracting any common sample center leaves its centered blocks unchanged.

[F6]

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 inherited from the sampling and conditional-block suppliers, including the latter's complete orthonormal basis; no further choice principle is added.

Proof

Given: The assumptions and suppliers above.

1.1givenF1F3F4F5F6A1algebra

Set $c_{b, ℓ} = a_{b} w ((ℓ - 1 / 2) / b)$ and extend this finite sequence by zero on all other integer indices. The exact normalization gives $b^{- 1} \sum_{ℓ} c_{b, ℓ}^{2} = 1$ , and [F4] bounds its maximum by a constant $A$ for sufficiently large $b$ . Symmetry and monotonicity of $w$ make the sequence nonnegative and unimodal, including the possible equal central pair; hence its discrete total variation, including both exterior zero tails, is at most $2 A$ . For every fixed integer $h \geq 1$ , telescoping $h$ successive shifts gives $\sum_{ℓ \in Z} ∣ c_{b, ℓ + h} - c_{b, ℓ} ∣ \leq 2 A h$ . Therefore $∣ b^{- 1} \sum_{ℓ = 1}^{b - h} c_{b, ℓ} c_{b, ℓ + h} - 1 ∣ \leq \frac{A}{b} \sum_{ℓ \in Z} ∣ c_{b, ℓ + h} - c_{b, ℓ} ∣ \leq \frac{2 A^{2} h}{b} ⟶ 0.$ This verifies every weight hypothesis of [F5], even for discontinuous monotone tapers. By [F6], the process satisfies the replacement-error rate needed by [F5].

2.1F3F5step 1.1algebra

For each full or partial weighted block, replacing $Y_{t}$ by $Y_{t} - {\overset{ˉ}{Y}}_{n}$ subtracts $(\sum_{ℓ = 1}^{q} c_{b, ℓ}) {\overset{ˉ}{Y}}_{n}$ from every candidate block with $q$ entries. Its empirical conditional center over the $N = n - b + 1$ starts subtracts the same vector, so all conditionally centered block sums agree exactly with those formed from $Y_{t}$ . Thus $\sqrt{n} ({\overset{ˉ}{Y}}_{n}^{*} - E^{*} {\overset{ˉ}{Y}}_{n}^{*})$ is precisely $Z_{n}^{*}$ in [F5], with the full-taper prefix weights on its last $r = n - b ⌊ n / b ⌋$ entries.

3.1F2F3F5step 1.1step 2.1∎

Apply [F5] with the sampling-law Gaussian $G$ supplied by [F2]. It gives conditional weak convergence to that same $G$ , conditional tightness, and convergence of the exact covariance $n {Cov}^{*} ({\overset{ˉ}{Y}}_{n}^{*})$ in trace norm, which implies its stated Hilbert-Schmidt convergence. The final prefix has expected conditional squared norm $O (b / n)$ and is negligible without requiring $b ∣ n$ . Its limiting covariance $Γ$ is the sampling target in [F2], completing both claims.

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

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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 2.2 (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)