Variance of Quadrature over Scrambled Unions of Nets

This article studies the variance of quadrature over a scrambled union of two nets, ( 0 ; 0; m; s)-net and ( 1 ; 0; m \Gamma d; s)-net in base b. For every integrand in L 2 [0; 1) s , a variance formula is given. Under some mild conditions on integrand, the rates of convergence are found. For smooth integrands, if d=m ! c with c ! 1 and d \Gamma cm is bounded as m ! 1, then the variance is of order n \Gamma3+c (log n) s\Gamma1 as n = 0 b m + 1 b m\Gammad !1. Key words and phrases: Integration, multiresolution, Quasi-Monte Carlo 1 Introduction We consider the problem of approximating integral I = Z [0;1) s f(x)dx for s 1 by the sample mean I = 1 n n X i=1 f(x i ); where x 1 ; : : : ; x n are n points in [0; 1) s . We assume that f is square integrable, that is, f 2 L 2 [0; 1) s . For high dimensional case, Monte Carlo methods and equidistribution or Quasi-Monte Carlo methods are most widely used. Under Monte Carlo methods, n points x i are drawn from th..