A discrepancy bound for deterministic acceptance-rejection samplers beyond $$N^{-1/2}$$N-1/2 in dimension 1

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 $$\\mathcal {O} (N^{-2/3}\\log N)$$O(N-2/3logN), provided that the target density is twice continuously differentiable with non-vanishing curvature and the AR sampler uses the driver sequence $$\\mathcal {K}_M= \\{( j \\alpha , j \\beta ) ~~ mod~~1 \\mid j = 1,\\ldots ,M\\},$$KM={(jź,jβ)mod1źj=1,ź,M}, where $$\\alpha ,\\beta $$ź,β are real algebraic numbers such that $$1,\\alpha ,\\beta $$1,ź,β is a basis of a number field over $$\\mathbb {Q}$$Q of degree 3. For the driver sequence $$\\mathcal {F}_k= \\{ ({j}/{F_k}, \\{{jF_{k-1}}/{F_k}\\} ) \\mid j=1,\\ldots , F_k\\},$$Fk={(j/Fk,{jFk-1/Fk})źj=1,ź,Fk}, where $$F_k$$Fk is the k-th Fibonacci number and $$\\{x\\}=x-\\lfloor x \\rfloor $${x}=x-źxź is the fractional part of a non-negative real number x, we can remove the $$\\log $$log factor to improve the convergence rate to $$\\mathcal {O}(N^{-2/3})$$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 $$\\mathcal {K}_M$$KM and $$\\mathcal {F}_k$$Fk as driver sequences. These results confirm that achieving a convergence rate beyond $$N^{-1/2}$$N-1/2 is possible in practice using $$\\mathcal {K}_M$$KM and $$\\mathcal {F}_k$$Fk as driver sequences in the acceptance-rejection sampler.