Discrepancy behaviour in the non-asymptotic regime

Abstract

We have computed true star-discrepancies D ∗(N) for Halton and Niederreiter point sequences in dimensions s=2, 3, 4 , and 6 with N up to 250 000 , 10 000 , 2000 and 300, respectively. For comparison, we also calculated some L 2 discrepancies T ∗ . The mean behaviour of D ∗(N) can well be approximated by power laws with an exponent between about −0.7 and −0.9, which slowly decreases with N. This behaviour is far from the generally assumed asymptotic one ∼ C( s)(log N) s / N. The factor between the true and the asymptotic behaviour increases strongly with s and reaches many orders of magnitude for large s. The ratios of D ∗(N) for different low discrepancy sequences are not proportional to the presumed asymptotic pre-factors C( s). Especially for the range of bases investigated, Niederreiter sequences have about the same D ∗ as Halton ones in a range of N, which we conjecture to be at least of the order of 10 10.