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Lindeberg–Lévy limit for the IID Monte Carlo mean

Statement

Assume the Axiom of Choice. Let $(Y_{i})_{i \geq 1}$ be a real IID sequence with $μ = E [Y_{1}]$ and $0 < σ^{2} : = Var (Y_{1}) < \infty$ . Then $\sqrt{n} ({\overset{ˉ}{Y}}_{n} - μ) \Rightarrow N (0, σ^{2}) .$

Facts & Assumptions

Given: AC, a real IID sequence $(Y_{i})_{i \geq 1}$ , and $μ = E [Y_{1}]$ with $0 < σ^{2} : = Var (Y_{1}) < \infty$ .

[A1]

The Axiom of Choice: Every family of nonempty sets has a choice function; the present theorem assumes this axiom.

[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: ${\overset{ˉ}{Y}}_{n} = n^{- 1} \sum_{i = 1}^{n} Y_{i}$ .

[F3]

Lindeberg-Levy iid central limit theorem: Assuming AC, for real IID variables with mean $m$ and variance $0 < σ^{2} < \infty$ , $(\sum_{i = 1}^{n} Y_{i} - n m) / (σ \sqrt{n}) \Rightarrow N (0, 1)$ .

[F4]

Continuous mapping theorem: If $X_{n} \Rightarrow X$ and $g$ is continuous, then $g (X_{n}) \Rightarrow g (X)$ .

[F5]

Standard normal and normal laws: Assuming AC, $N (m, σ^{2})$ is the law of $x \mapsto m + σ x$ under $N (0, 1)$ .

Proof

1.1A1F1F2F3givenalgebra

The hypotheses of [F3] hold by [A1], [F1], and the given positive finite variance. Since [F2] gives $\sum_{i = 1}^{n} Y_{i} = n {\overset{ˉ}{Y}}_{n}$ , [F3] yields $Z_{n} : = \sqrt{n} ({\overset{ˉ}{Y}}_{n} - μ) / σ \Rightarrow N (0, 1)$ .

2.1A1F4F5step 1.1algebra∎

The map $x \mapsto σ x$ is continuous because $σ > 0$ . Applying [F4] to step 1.1 gives $σ Z_{n} \Rightarrow σ Z$ for $Z \sim N (0, 1)$ . By [A1] and [F5], the law of $σ Z$ is $N (0, σ^{2})$ , and $σ Z_{n} = \sqrt{n} ({\overset{ˉ}{Y}}_{n} - μ)$ . This is the stated convergence.

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)