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.

Triangular taper weights for a block of length three

Example

Take the triangular window $w (u) = \max (1 - 2 ∣ u - \frac{1}{2} ∣, 0)$ and a block length $b = 3$ . The midpoint weights are $w_{3} = (1 / 3, 1, 1 / 3)$ .

Verification

Given: The window and block length in the Example.

[F1] Tapered block bootstrap: The sampled window weights are $w_{b} (r) = w ((r - \frac{1}{2}) / b)$ , with scale $a_{b} = \sqrt{b} / (\sum_{r} w_{b} (r)^{2})^{1 / 2}$ .

[F2] Asymptotic squared-norm normalization of a tapered window: For an admissible window, $W_{b}^{2} / b \to \int_{0}^{1} w (u)^{2} d u > 0$ ; the exact finite- $b$ scale is still computed from $W_{b}$ .

1.1F1givenalgebra

Substitution of $r = 1, 2, 3$ gives $w_{3} = (1 / 3, 1, 1 / 3)$ , so $W_{3}^{2} = 1 / 9 + 1 + 1 / 9 = 11 / 9$ and $a_{3} = 3 \sqrt{3 / 11}$ .

2.1F1F2step 1.1algebra∎

The scaled within-block weights are therefore $(\sqrt{3 / 11}, 3 \sqrt{3 / 11}, \sqrt{3 / 11})$ . Their squared sum is $\frac{3 + 27 + 3}{11} = 3$ , equal to $b$ , as required by the normalization $a_{b}^{2} W_{b}^{2} = b$ .

Depends on

Used by

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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, Assumption 2 (standard reference, not scraped)