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Marginal uniformity makes an RQMC rule unbiased

Statement

Let $X_{1}, \dots, X_{n}$ be a marginally uniform randomized point array in $[0, 1)^{d}$ , with no independence assumption among its points. For every Borel $f \in L^{1} ([0, 1)^{d})$ , the RQMC estimator ${\hat{I}}_{n} (f) = n^{- 1} \sum_{i = 1}^{n} f (X_{i})$ is integrable and $E [{\hat{I}}_{n} (f)] = \int_{[0, 1)^{d}} f (x) d x .$

Facts & Assumptions

Given: The probability space, the marginally uniform array, and a Borel integrable $f$ .

[F1]

Randomized quasi-Monte Carlo point sets and estimators: Each $X_{i}$ has normalized Lebesgue law, and ${\hat{I}}_{n} (f) = n^{- 1} \sum_{i} f (X_{i})$ .

[F2]

Change of variables for expectation: If $g (X)$ is integrable, then $E [g (X)] = \int g d P_{X}$ .

[F3]

The Lebesgue integral is linear on $L^{1} (μ)$ : The integral is linear on $L^{1}$ .

Proof

technique · direct

Given: $n \geq 1$ , $X_{1}, \dots, X_{n}$ with the marginal laws in [F1], and $f \in L^{1} ([0, 1)^{d})$ .

1.1F1F2given

By [F1], the law of every $X_{i}$ is normalized Lebesgue measure. Since $f$ is integrable for that law, [F2] gives $E [f (X_{i})] = \int_{[0, 1)^{d}} f (x) d x = I (f)$ for each $i$ , and each $f (X_{i})$ is integrable.

2.1F1F3step 1.1algebra∎

The finite sum of the integrable variables $f (X_{i})$ is integrable. By linearity, $E [{\hat{I}}_{n} (f)] = n^{- 1} \sum_{i = 1}^{n} E [f (X_{i})] = n^{- 1} \sum_{i = 1}^{n} I (f) = I (f)$ . No covariance or independence assumption is used.

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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