  We consider an acceptance-rejection sampler based on a deterministic driver
sequence. The deterministic sequence is chosen such that the discrepancy
between the empirical target distribution and the target distribution is small.
We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical
evidence shows convergence rates beyond the crude Monte Carlo rate of
$N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC
acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound
shows that for any given driver sequence, there always exists a target density
such that the star discrepancy is at most $N^{-2/(s+1)}$. 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
$\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced
acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F.
Barekat and R.Caflisch. Simulation with Fluctuation and Singular Rates.
ArXiv:1310.4555[math.NA], 2013.]
