ASYMPTOTIC NORMALITY OF SCRAMBLED GEOMETRIC NET QUADRATURE

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(logn)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 .d.