Laws of the Iterated Logarithm for Local Times of the Empirical Process

Abstract

We give exact expansions for the upper and lower tails of the distribution of the maximum of local time of standard Brownian bridge on interval [0, 1]. We use the above expansions to prove upper and lower laws of the iterated logarithm for the maximum of the local time of the uniform empirical process. This solves two open problems cited in the book of Shorack and Wellner.