Asymptotic Normality of Scrambled Geometric Net Quadrature

TL;DR

This paper proves that scrambled geometric net quadrature estimates are asymptotically normal for smooth functions, extending previous variance results and enhancing the theoretical understanding of this numerical integration method.

Contribution

It establishes the asymptotic normality of scrambled geometric net estimates, building on prior variance bounds and theoretical frameworks for smooth functions on product spaces.

Findings

Asymptotic normality of the estimator is proven for certain smooth functions.

Variance of the estimator is improved over traditional Monte Carlo methods.

Theoretical foundation for the distributional behavior of scrambled geometric nets.

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 $\mathbb{R}^d$.