ExampleConstruction: AI-generatedVerification: 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.

Gaussian integrand: exact Monte Carlo moments and asymptotic error

Example

Assume AC and let $X$ have law $N (μ, σ^{2})$ with $σ > 0$ . Use the identity integrand $g (x) = x$ , whose target is $E [g (X)] = μ$ , and take IID outputs $(Y_{i})_{i \geq 1}$ with this law. Then $E [{\overset{ˉ}{Y}}_{n}] = μ$ and $Var ({\overset{ˉ}{Y}}_{n}) = σ^{2} / n$ ; the estimated standard error is relatively consistent, and $\sqrt{n} ({\overset{ˉ}{Y}}_{n} - μ) \Rightarrow N (0, σ^{2})$ .

Facts & Assumptions

Given: AC, $σ > 0$ , and a real IID sequence $(Y_{i})_{i \geq 1}$ with common law $N (μ, σ^{2})$ .

[A1]

The Axiom of Choice: AC is the axiom that every family of nonempty sets has a choice function.

[F1]

Location-scale normal laws have their stated mean and variance: Under AC, a real random variable with law $N (m, σ^{2})$ for $σ \geq 0$ has mean $m$ and variance $σ^{2}$ .

[F2]

Unbiasedness and exact variance of the IID Monte Carlo mean: For integrable IID outputs, $E [{\overset{ˉ}{Y}}_{n}] = μ$ ; with a finite second moment, $Var ({\overset{ˉ}{Y}}_{n}) = σ^{2} / n$ .

[F3]

Consistency of the IID Monte Carlo sample variance: With finite second moment, $s_{n}^{2} \to σ^{2}$ almost surely, and for $σ^{2} > 0$ , ${\hat{s e}}_{n} / (σ / \sqrt{n}) \to 1$ almost surely.

[F4]

Lindeberg–Lévy limit for the IID Monte Carlo mean: Under AC and $0 < σ^{2} < \infty$ , $\sqrt{n} ({\overset{ˉ}{Y}}_{n} - μ) \Rightarrow N (0, σ^{2})$ .

[F5]

IID Monte Carlo sample mean, variance, and standard error: ${\hat{s e}}_{n} = s_{n} / \sqrt{n}$ for $n \geq 2$ under the finite-second-moment contract.

Verification

1.1A1F1F2given

By [F1], each output has mean $μ$ and variance $σ^{2}$ , so its first absolute moment and second moment are finite. Applying [F2] gives $E [{\overset{ˉ}{Y}}_{n}] = μ$ and $Var ({\overset{ˉ}{Y}}_{n}) = σ^{2} / n$ .

1.2A1F1F3F5given

The finite-second-moment hypotheses of [F3] hold by [A1] and [F1]. Thus $s_{n}^{2} \to σ^{2}$ almost surely and, using the plug-in definition [F5], ${\hat{s e}}_{n} / (σ / \sqrt{n}) \to 1$ .

2.1A1F1F4given∎

The sequence has positive finite variance and AC is assumed, so [F4] applies and gives $\sqrt{n} ({\overset{ˉ}{Y}}_{n} - μ) \Rightarrow N (0, σ^{2})$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 2 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed. (standard reference, not scraped)