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

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Unbiasedness and exact variance of the IID Monte Carlo mean

Statement

Let $(Y_{i})_{i \geq 1}$ be a real IID sequence. If $E ∣ Y_{1} ∣ < \infty$ and $μ = E [Y_{1}]$ , then $E [{\overset{ˉ}{Y}}_{n}] = μ$ for every $n \geq 1$ . If also $E [Y_{1}^{2}] < \infty$ and $σ^{2} = Var (Y_{1})$ , then $Var ({\overset{ˉ}{Y}}_{n}) = σ^{2} / n$ for every $n \geq 1$ .

Facts & Assumptions

Given: A real IID sequence $(Y_{i})_{i \geq 1}$ on one probability space; in the first claim $E ∣ Y_{1} ∣ < \infty$ , and in the second claim $E [Y_{1}^{2}] < \infty$ .

[F1]

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

[F2]

Change of variables for expectation: For a measurable $g$ , if $g (X)$ is integrable, then its expectation is the integral of $g$ against the law of $X$ .

[F3]

Linearity, monotonicity, and the modulus bound for expectation: Expectation is linear on integrable real random variables.

[F4]

Moments, variance, and covariance on a probability space: For square-integrable real $X$ , $Var (X) = E [(X - E [X])^{2}]$ .

[F5]

Variance adds for every finite pairwise-independent family: For a finite pairwise-independent family of square-integrable real random variables, the variance of the sum is the sum of the variances.

[F6]

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

Proof

1.1F1F2F4givenalgebra

Every $Y_{i}$ has the law of $Y_{1}$ by [F1]. The identity and $q (y) = (y - μ)^{2}$ are continuous polynomial maps by the elementary continuity rules for sums and products; inverse images of open sets under continuous maps are open, and inverse images preserve complements and countable unions, so both maps are Borel. Applying [F2] to the identity gives $E [Y_{i}] = μ$ when $E ∣ Y_{1} ∣ < \infty$ . If $E [Y_{1}^{2}] < \infty$ , applying [F2] to $q$ also gives $Var (Y_{i}) = σ^{2}$ by [F4].

2.1F3F6step 1.1algebra

By [F6], ${\overset{ˉ}{Y}}_{n} = n^{- 1} \sum_{i = 1}^{n} Y_{i}$ . Linearity [F3] and step 1.1 therefore give $E [{\overset{ˉ}{Y}}_{n}] = n^{- 1} \sum_{i = 1}^{n} E [Y_{i}] = μ$ .

3.1F1F3F4F5step 1.1givenalgebra∎

Assume $E [Y_{1}^{2}] < \infty$ . Since $∣ y ∣ \leq 1 + y^{2}$ for every real $y$ , each $Y_{i}$ is integrable as well as square-integrable. The first $n$ terms are pairwise independent by [F1], so [F5] and step 1.1 give $Var (\sum_{i = 1}^{n} Y_{i}) = n σ^{2}$ . Linearity [F3] and the variance definition [F4] give $Var (n^{- 1} \sum_{i = 1}^{n} Y_{i}) = n^{- 2} Var (\sum_{i = 1}^{n} Y_{i})$ ; hence $Var ({\overset{ˉ}{Y}}_{n}) = n^{- 2} (n σ^{2}) = σ^{2} / n$ .

Depends on

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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)