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Studentized RQMC replicate means are asymptotically normal as $R \to \infty$

Statement

Assume AC. Fix the data, the RQMC rule, and an integrable target $I = I (f)$ . Let ${\hat{I}}_{1}, {\hat{I}}_{2}, \dots$ be independent full-set randomizations of the same rule, with common mean $I$ and variance $0 < τ^{2} < \infty$ . Define ${\overline{I}}_{R} = R^{- 1} \sum_{r = 1}^{R} {\hat{I}}_{r}$ for $R \geq 2$ , and ${\hat{V}}_{R} = [R (R - 1)]^{- 1} \sum_{r = 1}^{R} ({\hat{I}}_{r} - {\overline{I}}_{R})^{2}$ . With the quotient assigned any fixed value when ${\hat{V}}_{R} = 0$ , $\frac{{\overline{I}}_{R} - I}{\sqrt{{\hat{V}}_{R}}} \Rightarrow N (0, 1) as R \to \infty .$ This limit is over independent randomizations for a fixed integrand and target. It asserts neither normality of a single scrambled-rule estimate nor accurate finite- $R$ Student interval coverage.

Facts & Assumptions

Given: AC, fixed data and rule, independent full-set RQMC estimates with mean $I$ and variance $0 < τ^{2} < \infty$ .

[F1]

Randomized quasi-Monte Carlo point sets and estimators: Full-set randomizations are copies of the entire point array, independent across replicates.

[F2]

Marginal uniformity makes an RQMC rule unbiased: Each marginally uniform full-set estimate has mean $I (f)$ .

[F3]

Independent RQMC replicates estimate randomization variance: ${\hat{V}}_{R}$ is the variance estimate for the replicate mean, with ${\hat{V}}_{R} = s_{R}^{2} / R$ for the usual sample variance $s_{R}^{2}$ .

[F4]

CLT for independent computation replicates: Under AC, IID replicate estimates with positive finite variance satisfy $\sqrt{R} ({\overline{I}}_{R} - I) / τ \Rightarrow N (0, 1)$ .

[F5]

Consistency of the IID Monte Carlo sample variance: For IID real variables with finite second moment, $s_{R}^{2} \to τ^{2}$ almost surely.

[F6]

Weak convergence of borel probability measures: The symbol $\Rightarrow$ denotes convergence against bounded continuous functions.

[F7]

Slutsky's theorem for real random variables: If $Y_{R} \Rightarrow Y$ and $W_{R} \to c \neq 0$ in probability, then $Y_{R} / W_{R} \Rightarrow Y / c$ , with an arbitrary fixed value assigned on $W_{R} = 0$ .

[A1]

The Axiom of Choice: AC is the stated axiom hypothesis passed to [F4].

Proof

technique · direct

Given: The assumptions and notation of the Statement.

1.1A1F1F2F4given

By [F1] the replicate estimates form an IID sequence. Their common mean is $I$ by [F2], and their variance is the positive finite number $τ^{2}$ by hypothesis. Under [A1], apply [F4] to obtain $Y_{R} : = \sqrt{R} ({\overline{I}}_{R} - I) / τ \Rightarrow N (0, 1)$ .

2.1F3F5step 1.1algebra

Let $s_{R}^{2} = (R - 1)^{- 1} \sum_{r = 1}^{R} ({\hat{I}}_{r} - {\overline{I}}_{R})^{2}$ . The finite second moment and [F5] give $s_{R}^{2} \to τ^{2}$ almost surely, hence $s_{R} / τ \to 1$ in probability. By [F3], $\sqrt{{\hat{V}}_{R}} = s_{R} / \sqrt{R}$ ; on $s_{R} > 0$ the displayed statistic equals $Y_{R} / (s_{R} / τ)$ .

3.1F6F7step 1.1step 2.1∎

Since $τ > 0$ , step 2.1 implies $P (s_{R} = 0) \to 0$ . Assigning any fixed value on that event matches the quotient convention in [F7]. Apply [F7] to step 1.1 with denominator $s_{R} / τ \to 1$ to obtain the stated standard-normal limit. The limit is in $R$ at fixed $n$ , not in the point count $n$ .

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Sources

Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §§17.1 and 17.4 (standard reference, not scraped)
Pierre L’Ecuyer and Christiane Lemieux, Recent Advances in Randomized Quasi-Monte Carlo Methods, §5 (replication context) (standard reference, not scraped)

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