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Variance bound for a nested scramble of a $(0, m, d)$ -net

Statement

Assume Countable Choice. Let $a_{1}, \dots, a_{n}$ be a $(0, m, d)$ -net in base $b \geq \max (d, 2)$ , where $n = b^{m}$ , and let $X_{1}, \dots, X_{n}$ be its nested uniform scramble. For real Borel $f \in L^{2} ([0, 1)^{d})$ , with $σ^{2} = Var (f (U))$ for $U \sim Unif ([0, 1)^{d})$ , $Var (\frac{1}{n} \sum_{i = 1}^{n} f (X_{i})) \leq {(\frac{b}{b - 1})}^{\min (m, d - 1)} \frac{σ^{2}}{n} \leq e \frac{σ^{2}}{n} .$

Facts & Assumptions

Given: Countable Choice, the stated net and scramble, and a real Borel square-integrable $f$ .

[F1]

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

[F2]

Sharp gain bound for scrambled $(0, m, d)$ -nets: For this net, every gain is at most $(b / (b - 1))^{\min (m, d - 1)}$ .

[F3]

Orthogonal base- $b$ multiresolution and variance decomposition: The component variances sum to $Var (f (U))$ .

[F4]

Moments, variance, and covariance on a probability space: $σ^{2}$ is finite and equals the centered second moment.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited through [F1] and [F3].

Proof

technique · direct

Given: The assumptions and notation in the Statement.

1.1A1F1F2

By [F1], the variance is $n^{- 1} \sum_{u, κ} Γ_{u, κ} σ_{u, κ}^{2} (f)$ . Each component variance is nonnegative, and [F2] bounds each gain above by $(b / (b - 1))^{\min (m, d - 1)}$ . Therefore the sum is at most $n^{- 1} (b / (b - 1))^{\min (m, d - 1)} \sum_{u, κ} σ_{u, κ}^{2} (f)$ .

2.1A1F3F4step 1.1algebra∎

By [F3] the component sum in step 1.1 is $σ^{2}$ . Also $\log ((b / (b - 1))^{\min (m, d - 1)}) = \min (m, d - 1) \log (1 + 1 / (b - 1)) \leq (d - 1) / (b - 1) \leq 1$ , because $\log (1 + x) \leq x$ for $x \geq 0$ and $b \geq d$ . Exponentiating gives the second inequality with $e$ .

Depends on

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Sources

Art B. Owen, Monte Carlo Variance of Scrambled Net Quadrature, Theorem 3 and proof, pp. 1902–1904 (standard reference, not scraped)
Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.5, Theorem 17.3 (standard reference, not scraped)

Variance bound for a nested scramble of a (0,m,d)(0,m,d)-net