Gaussian limits for discrepancies I. Asymptotic results

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 → ∞. We then examine the circumstances under which this distribution approaches a normal distribution. For large classes of non-uniformity measures, a ‘Central Limit Theorem’ can be derived.