On the convergence rate of Quasi Monte Carlo method with importance sampling for unbounded functions in RKHS

Abstract

Importance Sampling (IS), a variance reduction technique of significant efficacy in Monte Carlo (MC) simulation, is frequently utilized for Bayesian inference and other statistical challenges. Quasi-Monte Carlo (QMC) replaces the random samples in MC with low discrepancy points and has the potential to substantially enhance error rates. In this paper, we integrate IS with a randomly shifted rank-1 lattice rule, a widely used QMC method. Within the framework of Reproducing Kernel Hilbert spaces (RKHS), we establish the convergence rate of the lattice rule for a class of exponential growth unbounded integrands. Besides, we give the convergence order of IS combined with QMC on this class, which provides a reference for us to choose the importance density later.