TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Strong and weak consistency of the IID Monte Carlo mean

Statement

Let $(Y_{i})_{i \geq 1}$ be a real IID sequence with $E ∣ Y_{1} ∣ < \infty$ , and set $μ = E [Y_{1}]$ . Then $E ∣ {\overset{ˉ}{Y}}_{n} - μ ∣ \to 0$ , so ${\overset{ˉ}{Y}}_{n} \to μ$ in probability; moreover, ${\overset{ˉ}{Y}}_{n} \to μ$ almost surely.

Facts & Assumptions

Given: A real IID sequence $(Y_{i})_{i \geq 1}$ on one probability space with $E ∣ Y_{1} ∣ < \infty$ .

[F1]

Identical distribution and IID families: IID means that all variables have the same law and the whole family is mutually independent.

[F2]

IID Monte Carlo sample mean, variance, and standard error: The Monte Carlo sample mean is ${\overset{ˉ}{Y}}_{n} = n^{- 1} \sum_{i = 1}^{n} Y_{i}$ .

[F3]

Khinchin weak law for integrable IID variables: For real IID variables with finite first absolute moment, $E ∣ S_{n} / n - μ ∣ \to 0$ , and consequently $S_{n} / n \to μ$ in probability.

[F4]

Kolmogorov iid l1 strong law: For real IID variables with finite first absolute moment, $S_{n} / n \to μ$ almost surely.

Proof

1.1F1F2F3given

By [F2], ${\overset{ˉ}{Y}}_{n} = S_{n} / n$ . The hypotheses of [F3] hold by [F1] and the given integrability, so $E ∣ {\overset{ˉ}{Y}}_{n} - μ ∣ \to 0$ and ${\overset{ˉ}{Y}}_{n} \to μ$ in probability.

2.1F2F4given∎

Under the same hypotheses, [F4] gives $S_{n} / n \to μ$ almost surely. Rewriting with [F2] yields ${\overset{ˉ}{Y}}_{n} \to μ$ almost surely.

Depends on

Used by

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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)