Statistical Properties of Generalized Discrepancies and Related Quantities

When testing that a sample of n points in the unit hypercube [0, 1]d comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell (1996, 1998) introduced the so-called generalized Lp−discrepancies. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness of fit tests. The aim of this paper is to derive the strong and weak asymptotic properties of these statistics.