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

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Exact conditional covariance of a tapered block bootstrap mean

Statement

Fix a positive integer $n$ , observations $Y_{1}, \dots, Y_{n} \in H$ , an integer block length $b$ with $1 \leq b \leq n$ and $W_{b} > 0$ , and the taper and scale $w_{b}, a_{b}$ from Tapered block bootstrap. Set $R_{t} = Y_{t} - {\overset{ˉ}{Y}}_{n}$ , $N = n - b + 1$ , $k = ⌊ n / b ⌋$ , and $r = n - k b$ . For $q = b$ , and also $q = r$ when $r > 0$ , define the tapered partial block sum

$T_{j}^{(q)} = a_{b} \sum_{ℓ = 1}^{q} w_{b} (ℓ) R_{j + ℓ - 1}, {\overset{ˉ}{T}}_{q} = N^{- 1} \sum_{j = 1}^{N} T_{j}^{(q)}, V_{q} = N^{- 1} \sum_{j = 1}^{N} (T_{j}^{(q)} - {\overset{ˉ}{T}}_{q}) \otimes (T_{j}^{(q)} - {\overset{ˉ}{T}}_{q}) .$

Set $V_{0} = 0$ . If ${\overset{ˉ}{X}}_{n}^{*}$ is the tapered bootstrap mean, then

$n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*}) = \frac{k V_{b} + V_{r}}{n},$

with the $V_{r}$ term zero when $r = 0$ .

Facts & Assumptions

Given: Fixed observations, a positive taper norm $W_{b}$ , and the conditional block-index law in the Definition.

[F1]

Tapered block bootstrap: Each resampled block consists of the centered observations multiplied positionwise by $a_{b} w_{b} (ℓ)$ ; the starts are independent uniform indices, and the final block is truncated if necessary.

Proof

Given: The data and bootstrap randomization in the Statement.

1.1F1givenalgebra

Write $n = k b + r$ , $0 \leq r < b$ . The sum of the first $n$ tapered bootstrap observations is the sum of $k$ complete tapered block sums and, if $r > 0$ , the first $r$ tapered entries of one additional block. Conditional on the data these contributions are independent; their centered covariance operators are $V_{b}$ and, when present, $V_{r}$ , because every block start is uniform on the same $N$ choices.

2.1step 1.1algebra∎

Divide the block-sum total by $n$ . Independence and centering eliminate all off-diagonal tensor terms, so ${Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*}) = n^{- 2} (k V_{b} + V_{r})$ . Multiplication by $n$ proves the identity.

Depends on

Tapered block bootstrap

Used by

Tapered block bootstrap CLT for dependent Hilbert-valued data Theorem

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), Section 2.3 (standard reference, not scraped)