On the Convergence Rate of Randomized Quasi--Monte Carlo for Discontinuous Functions

Abstract

This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only $o(n^{-1/2})$ for discontinuous functions. For certain discontinuous functions in $d$ dimensions, we prove that the RMSE of RQMC is $O(n^{-1/2-1/(4d-2)+\epsilon})$ for any $\epsilon>0$ and arbitrary $n$. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to $O(n^{-1/2-1/(4d_u-2)+\epsilon})$, where $d_u$ denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is $O(n^{-1/2-1/(2d)})$ for certain indicator functions.