  In this paper we propose an acceptance-rejection sampler using stratified
inputs as diver sequence. We estimate the discrepancy of the points generated
by this algorithm. First we show an upper bound on the star discrepancy of
order $N^{-1/2-1/(2s)}$. Further we prove an upper bound on the $q$-th moment
of the $L_q$-discrepancy $(\mathbb{E}[N^{q}L^{q}_{q,N}])^{1/q}$ for $2\le q\le
\infty$, which is of order $N^{(1-1/s)(1-1/q)}$. We also present an improved
convergence rate for a deterministic acceptance-rejection algorithm using
$(t,m,s)-$nets as driver sequence.
