Asymptotic Normality of Scrambled Geometric Net Quadrature

Abstract

In a very recent work, Basu and Owen (2015) 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 develop on the work by Loh (2003), to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of $\mathbb{R}^d$.