ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 finite moving-block bootstrap calculation

Example

Apply Moving block bootstrap to the scalar data $0, 1, 2, 3$ with block length $b = 2$ . There are three overlapping blocks, $(0, 1), (1, 2), (2, 3)$ , and two independent uniform block starts are drawn and concatenated.

Verification

Given: The four data values and block length in the Example.

[F1] Moving block bootstrap: A length-two moving block starts at one of the three positions uniformly; two such starts are sampled independently and the concatenation has length four.

1.1F1givenalgebra

At the first position of a selected block, the conditional average over starts is $(0 + 1 + 2) / 3 = 1$ . At the second position it is $(1 + 2 + 3) / 3 = 2$ .

2.1step 1.1givenalgebra∎

The output mean is the average of two first-position values and two second-position values, so its conditional expectation is $(1 + 2) / 2 = 3 / 2$ , which equals the observed sample mean. The resampled blocks preserve each adjacent pair inside its chosen block; only the join between blocks can alter adjacency.

Depends on

Moving block bootstrap

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)