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

We consider the approximation of d-dimensional weighted integrals of certain isotropic functions. We are mainly interested in cases where d is large. We show that the convergence rate of quasi-Monte Carlo for the approximation of these integrals is O( √ logn/n). Since this is a worst case result, compared to the expected convergence rate 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 n/n), of quasi-Monte Carlo.