Discrepancy behaviour in the non-asymptotic regime

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 similar to C(s)(Iog 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). (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.