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 $\mathcal {K}_M= \{( j \alpha , j \beta ) ~~ mod~~1 \mid j = 1,\ldots ,M\},$ where $\alpha ,\beta$ are real algebraic numbers such that $1,\alpha ,\beta$ 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.