General discrepancy estimates: the Walsh function system

(A) generation of uniform pseudorandom numbers (in the normalized domain [0, 1[), (B) quasi-Monte Carlo methods (i.e. random samples in a Monte Carlo method are replaced by deterministic points) a well-chosen finite point set P = {x0,x1, . . . ,xN−1} in the s-dimensional unit cube [0, 1[ has to be generated. To assess the quality of P, it is essential to determine the deviation of the (empirical) distribution of P from uniform distribution on [0, 1[ (see Niederreiter [5, Chapters 2 and 7] for a thorough discussion). Discrepancy has turned out to be the appropriate concept to measure this deviation. There are several notions of discrepancy. The most important are the following.