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In this paper we consider an acceptance-rejection (AR) sampler based ondeterministic driver sequences. We prove that the discrepancy of an $N$ elementsample set generated in this way is bounded by $\mathcal{O} (N^{-2/3}\log N)$,provided that the target density is twice continuously differentiable withnon-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 abasis of a number field over $\mathbb{Q}$ of degree $3$. For the driversequence $$\mathcal{F}_k= \{ ({j}/{F_k}, \{{jF_{k-1}}/{F_k}\} ) \midj=1,\ldots, F_k\},$$ where $F_k$ is the $k$-th Fibonacci number and$\{x\}=x-\lfloor x \rfloor$ is the fractional part of a non-negative realnumber $x$, 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-discrepancyof samples generated for some target densities using $\mathcal{K}_M$ and$\mathcal{F}_k$ as driver sequences. These results confirm that achieving aconvergence rate beyond $N^{-1/2}$ is possible in practice using$\mathcal{K}_M$ and $\mathcal{F}_k$ as driver sequences in theacceptance-rejection sampler.

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