Gaussian limits for discrepancies. I: Asymptotic results

Abstract

We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit $N\to \infty$. We then examine the circumstances under which this distribution approaches a normal distribution. For large classes of non-uniformity measures, a Law of Many Modes in the spirit of the Central Limit Theorem can be derived.

Comments: 25 pages, Latex, uses fleqn.sty, a4wide.sty, amsmath.sty