CorollaryStatement: 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.

CLT for independent computation replicates

Statement

Assume AC. Let $(Z_{r})_{r \geq 1}$ be IID real computation-replicate outputs with mean $μ$ and variance $0 < τ^{2} < \infty$ . Then, as $R \to \infty$ , $\frac{\sqrt{R} ({\overline{Z}}_{R} - μ)}{τ} \Rightarrow N (0, 1), {\overline{Z}}_{R} : = \frac{1}{R} \sum_{r = 1}^{R} Z_{r} .$ This is a limit for the replicate-randomization law under the stated fixed-target setup; it is not a data-sampling confidence statement and does not by itself justify a finite- $R$ Student interval.

Facts & Assumptions

Given: Assume AC. Let $(Z_{r})_{r \geq 1}$ be IID real random variables with mean $μ$ and variance $0 < τ^{2} < \infty$ . Prove the standardized replicate-mean convergence.

[F1]

IID means the family is independent and identically distributed. (Identical distribution and IID families)

[F2]

For a real random variable with finite second moment, variance is $E [(Z - E Z)^{2}]$ . (Moments, variance, and covariance on a probability space)

[F3]

Under AC, IID real variables with finite positive variance satisfy the Lindeberg–Levy normalized-sum convergence. (Lindeberg-Levy iid central limit theorem)

[A1]

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

Proof

1.1givenF1F2algebra

The normalized replicate average is $\sqrt{R} ({\overline{Z}}_{R} - μ) / τ = (\sum_{r = 1}^{R} Z_{r} - R μ) / (τ \sqrt{R})$ , the normalized sum in the Lindeberg–Levy theorem.

2.1givenA1F3step 1.1∎

Under [A1], apply [F3] to the IID sequence in step 1.1 to obtain the asserted convergence in distribution.

Depends on

Used by

Studentized RQMC replicate means are asymptotically normal as R→∞ Corollary

Dependency tree · two levels

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Sources

Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.1 and end notes (standard reference, not scraped)