Computational aspects of discrepancies for equidistribution on the hypercube

In this paper, we study the asymptotic statistical properties of some discrepancies defined on the unit hypercube, originally introduced in Numerical Analysis to assess the equidistribution of low-discrepancy sequences. We show that they have highly desirable properties. Nevertheless, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. This raises some problems concerning the approximation of the asymptotic distribution. These issues are considered in detail: several solutions are proposed and compared, and bounds for the approximation error are discussed.