The Asymptotic Distribution of Quadratic Discrepancies

Summary In Numerical Analysis, several discrepancies have been introduced to test that a sample of n points in the unit hypercube [0, 1]d comes from a uniform distribution. An outstanding example is given by Hickernell’s generalized $$\mathcal{L}^P $$ -discrepancies, that constitute a generalization of the Kolmogorov-Smirnov and the Cramér-von Mises statistics. These discrepancies can be used in numerical integration by Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity and goodness of fit tests. In this paper, after having recalled some necessary asymptotic results derived in companion papers, we show that the case of $$\mathcal{L}^2 $$ -discrepancies is more convenient to handle and we provide a new computational approximation of their asymptotic distribution. As an illustration, we show that our algorithm is able to recover the tabulated asymptotic distribution of the Cramér-von Mises statistic. The results so obtained are very general and can be applied with minor modifications to other discrepancies, such as the diaphony, the weighted spectral test, the Fourier discrepancy and the class of chi-square tests.