ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

A block-length sequence in the bootstrap CLT regime

Example

For $b_{n} = ⌈ n^{1 / 3} ⌉$ , the two block-length conditions in the moving and tapered block bootstrap theorems are satisfied.

Verification

Given: $b_{n} = ⌈ n^{1 / 3} ⌉$ and the theorem regime $b_{n} \to \infty$ , $b_{n} / \sqrt{n} \to 0$ .

[F1] Overlapping block-index resampling: A block length is an integer $b_{n}$ between $1$ and $n$ .

1.1F1givenalgebra

Since $n^{1 / 3} \leq b_{n} < n^{1 / 3} + 1$ , $b_{n} \to \infty$ , and $b_{n}$ is a positive integer. For every integer $n \geq 1$ , $n^{1 / 3} \leq n$ ; since $n$ itself is an integer, taking the least integer above $n^{1 / 3}$ gives $b_{n} \leq n$ . Thus the block length is admissible at every positive sample size.

2.1givenstep 1.1algebra∎

Dividing the same bounds by $\sqrt{n}$ gives $0 \leq b_{n} / \sqrt{n} < n^{- 1 / 6} + n^{- 1 / 2} \to 0$ . Thus $b_{n}^{- 1} + b_{n} n^{- 1 / 2} \to 0$ .

Depends on

Overlapping block-index resampling

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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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), Theorems 2.1 and 2.2 (standard reference, not scraped)