Discrepancy Estimates for Acceptance-Rejection Samplers Using Stratified Inputs

TL;DR

This paper introduces an acceptance-rejection sampling method using stratified inputs, providing theoretical bounds on discrepancy and convergence rates, and improves deterministic algorithms with $(t,m,s)$-nets.

Contribution

It offers new discrepancy bounds for stratified input-based acceptance-rejection samplers and enhances convergence rates for deterministic algorithms using $(t,m,s)$-nets.

Findings

Star discrepancy bound of order $N^{-1/2-1/(2s)}$

Upper bound on the $q$-th moment of $L_q$-discrepancy of order $N^{(1-1/s)(1-1/q)}$

Improved convergence rate for deterministic algorithms with $(t,m,s)$-nets

Abstract

In this paper we propose an acceptance-rejection sampler using stratified inputs as diver sequence. We estimate the discrepancy of the points generated by this algorithm. First we show an upper bound on the star discrepancy of order $N^{-1/2-1/(2s)}$. Further we prove an upper bound on the $q$-th moment of the $L_q$-discrepancy $(\mathbb{E}[N^{q}L^{q}_{q,N}])^{1/q}$ for $2\le q\le \infty$, which is of order $N^{(1-1/s)(1-1/q)}$. We also present an improved convergence rate for a deterministic acceptance-rejection algorithm using $(t,m,s)-$nets as driver sequence.