TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Pathwise error decomposition along a reference chain

Statement

Let $θ = r_{0}, r_{1}, \dots, r_{m}, \tilde{r}$ be the finite scalar reference chain and returned value of Successive statistical and computational error targets, with $m \geq 0$ . Then, pointwise, $\tilde{r} - θ = \sum_{j = 1}^{m} (r_{j} - r_{j - 1}) + (\tilde{r} - r_{m}),$ and consequently $∣ \tilde{r} - θ ∣ \leq \sum_{j = 1}^{m} ∣ r_{j} - r_{j - 1} ∣ + ∣ \tilde{r} - r_{m} ∣ .$ The sum is empty when $m = 0$ . No independence, unbiasedness, or distributional assumption is needed.

Facts & Assumptions

Given: A finite chain $θ = r_{0}, r_{1}, \dots, r_{m}, \tilde{r}$ of real values, with $m \geq 0$ . Prove the stated identity and bound.

[F1]

The successive signed errors are $e_{j} = r_{j} - r_{j - 1}$ for $1 \leq j \leq m$ , the last numerical error is $e_{f p} = \tilde{r} - r_{m}$ , and $r_{0} = θ$ . (Successive statistical and computational error targets)

Proof

1.1givenF1algebra

By [F1], the finite sum telescopes: $\sum_{j = 1}^{m} (r_{j} - r_{j - 1}) = r_{m} - r_{0}$ ; for $m = 0$ both sides are $0$ .

2.1step 1.1F1algebra

Adding $\tilde{r} - r_{m}$ to the equality in step 1.1 and using $r_{0} = θ$ gives $\tilde{r} - θ = \sum_{j = 1}^{m} (r_{j} - r_{j - 1}) + (\tilde{r} - r_{m})$ .

3.1step 2.1algebra∎

Apply the triangle inequality to the finite sum in step 2.1 to obtain the absolute-error bound.

Depends on

Successive statistical and computational error targets

Used by

Dependency tree · two levels

4 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

Anthony Davison and David Hinkley, Bootstrap Methods and Their Application, Chapter 2 §2.5.1–§2.5.2 (standard reference, not scraped)
Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §§17.1 and 17.4 and end notes (standard reference, not scraped)