A Discrepancy Bound for a Deterministic Acceptance-Rejection Sampler

We consider an acceptance-rejection sampler based on a deterministic driver sequence, which is a simple case of a Markov chain quasi-Monte Carlo (MCQMC) method. The deterministic sequence is chosen such that the discrepancy between the empirical proposal distribution and the proposal distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler converges at a rate of $N^{-1/s}$ for a given target density defined on $[0,1]^{s-1}$. For some special cases we prove a convergence rate of $N^{-\alpha}$, where $1/s \le \alpha < 1$ depends on the target density. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)-$dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Our numerical experiments show convergence rates beyond the plain Monte Carlo rate of $N^{-1/2}$.