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 $O(\sqrt{\log n}/n)$. 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, $O(\log^d n/n)$, of quasi-Monte Carlo.