The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands

Self-normalized importance sampling (SNIS) is a fundamental tool in Bayesian inference when the posterior distribution involves an unknown normalizing constant. Although $L_1$-error (bias) and $L_2$-error (root mean square error) estimates of SNIS are well established for bounded integrands, results for unbounded integrands remain limited, especially under randomized quasi-Monte Carlo (RQMC) sampling. In this work, we derive $L_p$-error rate $(p\ge 1)$ for RQMC-based SNIS (RQMC-SNIS) estimators with unbounded integrands on unbounded domains. A key step in our analysis is to first establish the $L_p$-error rate for plain RQMC integration. Our results allow for a broader class of transport maps used to generate samples from RQMC points. Under mild function boundary growth conditions, we further establish $L_p$-error rate of order $\mathcal{O}(N^{-\beta +\epsilon })$ for RQMC-SNIS estimators, where $\epsilon >0$ is arbitrarily small, $N$ is the sample size, and $\beta \in (0,1]$ depends on the boundary growth rate of the resulting integrand. Numerical experiments validate the theoretical results.