Fast convergence of quasi-Monte Carlo for a class of isotropic integrals

Abstract

We consider the approximation of d d -dimensional weighted integrals of certain isotropic functions. We are mainly interested in cases where d d is large. We show that the convergence rate of quasi-Monte Carlo for the approximation of these integrals is O ( log ⁡ n / n ) O(\sqrt {\log n}/n) . Since this is a worst case result, compared to the expected convergence rate O ( n − 1 / 2 ) O(n^{-1/2}) of Monte Carlo, it shows the superiority of quasi-Monte Carlo for this type of integral. This is much faster than the worst case convergence, O ( log d ⁡ n / n ) O(\log ^d n/n) , of quasi-Monte Carlo.