ON G-DISCREPANCY AND MIXED MONTE CARLO AND QUASI-MONTE CARLO SEQUENCES

The G-star and the G-discrepancy are generalizations of the well known star and the extreme discrepancy; thereby G denotes a given con- tinuous distribution function on the d-dimensional unit cube (0,1) d . We list and prove some results that describe the behavior of the G-star and the G- discrepancy in terms of the dimension d and the number of sample points N. Our main focus is on so-called mixed sequences, which are d-dimensional hybrid-Monte Carlo sequences that result from concatenating a d 0 -dimensional deterministic sequence q with a d 00 -dimensional sequence X of independently G-distributed random vectors (here d = d 0 +d 00 ). We show that a probabilistic bound on the G-discrepancy of mixed sequence from (11) is unfortunately incor- rect, and correct it by proving a new probabilistic bound for the G-discrepancy of mixed sequences. Moreover this new bound exhibits for fixed dimension d a better asymptotical behavior in N, and a similar bound holds also for the G-star discrepancy.

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