A Discrepancy Bound for Deterministic Acceptance-Rejection Samplers Beyond $N^{-1/2}$ in Dimension 1

Abstract

In this paper we consider an acceptance-rejection (AR) sampler based on deterministic driver sequences. We prove that the discrepancy of an $N$ element sample set generated in this way is bounded by $\mathcal{O} (N^{-2/3}\log N)$, provided that the target density is twice continuously differentiable with non-vanishing curvature and the AR sampler uses the driver sequence $$\mathcal{K}_M= \{( j α, j β) ~~ mod~~1 \mid j = 1,\ldots,M\}, $$ where $α,β$ are real algebraic numbers such that $1,α,β$ is a basis of a number field over $\mathbb{Q}$ of degree $3$. For the driver sequence $$\mathcal{F}_k= \{ ({j}/{F_k}, \{jF_{k-1}/{F_k}\} ) \mid j=1,\ldots, F_k\},$$ where $F_k$ is the $k$-th Fibonacci number and $\{x\}=x-\lfloor x \rfloor$ is the fractional part of a non-negative real number $x$, we can remove the $\log$ factor to improve the convergence rate to $\mathcal{O}(N^{-2/3})$, where again $N$ is the number of samples we accepted. We also introduce a criterion for measuring the goodness of driver sequences. The proposed approach is numerically tested by calculating the star-discrepancy of samples generated for some target densities using $\mathcal{K}_M$ and $\mathcal{F}_k$ as driver sequences. These results confirm that achieving a convergence rate beyond $N^{-1/2}$ is possible in practice using $\mathcal{K}_M$ and $\mathcal{F}_k$ as driver sequences in the acceptance-rejection sampler.