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How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A smooth integrand satisfies the bound while an integrable singularity does not

Example

Assume Countable Choice. In one dimension compare $f_{1} (x) = x$ with the integrable singular function $f_{2} (x) = x^{- 3 / 4}$ for $x > 0$ and $f_{2} (0) = 0$ . The first satisfies the smooth scrambled-net variance hypotheses; the second does not even have a finite second moment.

Facts & Assumptions

Given: Countable Choice, a nested uniform scramble of a $(0, m, 1)$ -net in base $2$ , and the two integrands.

[F1]

Smooth-integrand variance rate for nestedly scrambled $(λ, t, m, d)$ -nets: A continuously differentiable one-dimensional integrand has scrambled-net variance $O (n^{- 3})$ for fixed base and net quality.

[F2]

Randomized quasi-Monte Carlo point sets and estimators: Each scrambled point is marginally uniform when the RQMC marginal condition holds.

[F3]

Expectation of a nonnegative or integrable random variable: Integrability and second moments are Lebesgue integrals against the uniform law.

[F4]

Nested uniform scrambles preserve net counts and uniform marginals: Each scrambled point has uniform law and the net property is preserved almost surely.

[F5]

Monotone convergence for the integral: Increasing nonnegative measurable truncations have integrals increasing to the integral of their pointwise limit.

[F6]

Change of variables for expectation: For a nonnegative Borel function $g$ and random element $X$ , $E [g (X)] = \int g d P_{X}$ .

[F7]

Moments, variance, and covariance on a probability space: Variance is defined for square-integrable variables and is finite only when their second moment is finite.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited from [F1].

Verification

1.1A1F1given

The function $f_{1} (x) = x$ has continuous derivative $1$ on $[0, 1]$ . With $n = 2^{m}$ , the hypotheses of [F1] hold, so its randomized-rule variance is $O (n^{- 3})$ .

1.2F3F5givenalgebra

For $N \geq 1$ , the nonnegative Borel truncations $g_{N} = f_{2} 1_{[1 / N, 1]}$ increase pointwise to $f_{2}$ and satisfy $\int_{0}^{1} g_{N} (x) d x = 4 (1 - N^{- 1 / 4})$ . Likewise, $h_{N} = f_{2}^{2} 1_{[1 / N, 1]}$ increase pointwise to $f_{2}^{2}$ and satisfy $\int_{0}^{1} h_{N} (x) d x = 2 (\sqrt{N} - 1)$ . By [F5], $\int_{0}^{1} f_{2} = 4$ while $\int_{0}^{1} f_{2}^{2} = + \infty$ ; $f_{2}$ is not continuously differentiable at $0$ .

2.1F2F4F6F7step 1.2algebra∎

By [F2], ${\hat{I}}_{n} (f_{2}) = n^{- 1} \sum_{i} f_{2} (X_{i})$ ; [F4] gives every point uniform law. Since $f_{2} \geq 0$ , [F6] and step 1.2 give $E [f_{2} (X_{1})^{2}] = + \infty$ . The squared estimate obeys ${\hat{I}}_{n} (f_{2})^{2} \geq n^{- 2} f_{2} (X_{1})^{2}$ , so its second moment is infinite. By [F7] it has no finite variance, and the smooth variance theorem does not apply to this integrable singularity.

Depends on

Used by

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Sources

Art B. Owen, Local Antithetic Sampling with Scrambled Nets, §5.1 (standard reference, not scraped)
Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.5 (standard reference, not scraped)