Asymptotic normality of scrambled geometric net quadrature

Abstract

In a very recent work, Basu and Owen [Found. Comput. Math. 17 (2017) 467–496] propose the use of scrambled geometric nets in numerical integration when the domain is a product of $s$ arbitrary spaces of dimension $d$ having a certain partitioning constraint. It was shown that for a class of smooth functions, the integral estimate has variance $O(n^{-1-2/d}(\log n)^{s-1})$ for scrambled geometric nets compared to $O(n^{-1})$ for ordinary Monte Carlo. The main idea of this paper is to expand on the work by Loh [Ann. Statist. 31 (2003) 1282–1324] to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of $\mathbb{R}^{d}$.