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Smooth-integrand variance rate for nestedly scrambled $(λ, t, m, d)$ -nets

Statement

Assume Countable Choice. Fix $b \geq 2$ , $d \geq 1$ , $t \geq 0$ , and $1 \leq λ < b$ . For each sufficiently large $m \geq t$ , let $a_{1}, \dots, a_{n}$ be a $(λ, t, m, d)$ -net in base $b$ , where $n = λ b^{m}$ , and let $X_{1}, \dots, X_{n}$ be its nested uniform scramble. Suppose $f$ is real, continuous on $[0, 1]^{d}$ , and every nonempty mixed coordinate derivative $\partial_{u} f$ exists and is continuous on the cube. Then $Var (\frac{1}{n} \sum_{i = 1}^{n} f (X_{i})) = O (n^{- 3} (\log n)^{d - 1}) (m \to \infty) .$ The implicit constant may depend on $b, d, t, λ$ and the suprema of the mixed derivatives; the limit is along $n = λ b^{m}$ .

Facts & Assumptions

Given: Countable Choice, fixed $b, d, t, λ$ , the nets and their nested scrambles, and the smooth integrand.

[F1]

ANOVA variance identity for nestedly scrambled point sets: The variance is the sum $n^{- 1} \sum_{u, κ} Γ_{u, κ} σ_{u, κ}^{2} (f)$ .

[F2]

Fixed-quality net gain coefficients are uniformly bounded: For fixed $b, t, d, λ$ , gains vanish if $∣ κ ∣ + ∣ u ∣ \leq m - t$ and otherwise $∣ Γ_{u, κ} ∣ \leq λ b^{t + d - 1}$ .

[F3]

Smoothness bounds the high-frequency ANOVA components: For each $u \neq \emptyset$ , $σ_{u, κ}^{2} (f) \leq C_{u} b^{- 2 ∣ κ ∣}$ with finite $C_{u}$ independent of $m$ .

[F4]

Moments, variance, and covariance on a probability space: The target variance is the centered second moment, finite under the stated continuity.

[F5]

For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: A continuous real function on the compact cube is bounded.

[A1]

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

Proof

technique · direct

Given: The objects and hypotheses of the Statement.

1.1A1F1F2F3F4F5

By [F5], $f$ is bounded on the compact cube and is square-integrable; [F4] identifies its variance as the finite centered second moment. Put $G = λ b^{t + d - 1}$ . By [F1]–[F3] and nonnegativity of the component variances, $Var ({\hat{I}}_{n}) \leq \frac{G}{n} \sum_{u \neq \emptyset} C_{u} \sum_{\begin{array}{c} κ \in N^{∣ u ∣} \\ ∣ κ ∣ > m - t - ∣ u ∣ \end{array}} b^{- 2 ∣ κ ∣}$ , where each $C_{u}$ is fixed independently of $m$ .

2.1step 1.1algebra

Fix $u$ and write $p = ∣ u ∣$ . The number of nonnegative $p$ -tuples with coordinate sum $r$ is $(\binom{r + p - 1}{p - 1})$ . For $m > t + d$ , reindexing with $s = r - (m - t - p) \geq 1$ gives $\sum_{∣ κ ∣ > m - t - p} b^{- 2 ∣ κ ∣} = b^{- 2 (m - t - p)} \sum_{s \geq 1} (\binom{m - t + s - 1}{p - 1}) b^{- 2 s} .$ For fixed $p, b, t$ , $(\binom{m - t + s - 1}{p - 1}) \leq (m + s)^{p - 1} / (p - 1)!$ ; for $m \geq 1$ , $(m + s)^{p - 1} \leq m^{p - 1} (1 + s)^{p - 1}$ . The series $\sum_{s \geq 1} (1 + s)^{p - 1} b^{- 2 s}$ converges, so the display is $O (b^{- 2 m} m^{p - 1}) = O (n^{- 2} (\log n)^{p - 1})$ , since $b^{- 2 m} = λ^{2} n^{- 2}$ and $m = \log_{b} (n / λ)$ .

3.1F3step 1.1step 2.1algebra∎

There are finitely many nonempty subsets $u$ , with $p \leq d$ . Substituting the bounds from step 2.1 into step 1.1 gives $Var ({\hat{I}}_{n}) = O (n^{- 3} (\log n)^{d - 1})$ . The constants depend only on the fixed parameters and the derivative bounds in [F3].

Depends on

Used by

A smooth integrand satisfies the bound while an integrable singularity does not Example

Dependency tree · two levels

32 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 §17.5, Theorem 17.5 (standard reference, not scraped)
Art B. Owen, Local Antithetic Sampling with Scrambled Nets, §5.1, Lemmas 2–3 and Theorem 3 (standard reference, not scraped)

Smooth-integrand variance rate for nestedly scrambled (λ,t,m,d)(λ,t,m,d)-nets