ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

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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.

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A monotone uniform integrand has a zero-variance antithetic pair

Example

Take $d = 1$ , $T (u) = u$ , $f (x) = x$ , and $U \sim Unif [0, 1]$ . The antithetic pair average is constant, while the ordinary mean of two independent draws has positive variance.

Facts & Assumptions

Given: $U$ uniform on $[0, 1]$ , $T (u) = u$ , and $f (x) = x$ .

[F1]

Antithetic randomized sampling: A pair averages $f (T (U))$ and $f (T (1 - U))$ .

[F2]

Antithetic variance and a monotone sufficient condition: The antithetic variance is $(σ^{2} + Cov (Y, \tilde{Y})) / (2 n)$ ; it is no larger than the IID comparator exactly when the covariance is nonpositive, and is strictly smaller exactly when the covariance is negative.

Verification

Given: The one-dimensional uniform input and identity sampler/integrand.

1.1givenalgebra

Direct integration gives $E U = E (1 - U) = 1 / 2$ , $Var (U) = 1 / 12$ , and $E [U (1 - U)] = 1 / 6$ , so $Cov (U, 1 - U) = 1 / 6 - 1 / 4 = - 1 / 12$ .

2.1F1F2step 1.1algebra∎

For each pair, $(U + (1 - U)) / 2 = 1 / 2$ , so [F1] gives a constant estimator of variance zero. By [F2], the comparator variance for two independent uniform draws is $(1 / 12) / 2 = 1 / 24$ , hence the reduction is strict.

Depends on

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