AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Overlapping block-index resampling

Definition

Input: a finite sequence $x_{1}, \dots, x_{n}$ in a measurable space, with $n \geq 1$ , and an integer block length $b$ satisfying $1 \leq b \leq n$ . Set $N = n - b + 1$ and $k = ⌈ n / b ⌉$ .

Randomization: draw $J_{1}, \dots, J_{k}$ independently and uniformly from ${1, \dots, N}$ .

Output: in order $s = 1, \dots, k$ , append the block $x_{J_{s}}, x_{J_{s} + 1}, \dots, x_{J_{s} + b - 1}$ , then retain the first $n$ entries of the concatenation. Thus the final selected block is truncated to its first $r = n - b ⌊ n / b ⌋$ entries when $r > 0$ . Conditional expectation given the input sequence means expectation over the indices $J_{s}$ alone.

Used by

Moving block bootstrap Algorithm
Null-constrained block bootstrap for independent series Algorithm
Tapered block bootstrap Algorithm
Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
A block-length sequence in the bootstrap CLT regime Example

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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 2.2 (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)