Functional Laws of the Iterated Logarithm for Local Empirical Processes Indexed by Sets

Abstract

We introduce the notion of a multivariate local (and tail) empirical process indexed by sets and establish a number of functionals laws of the iterated logarithm for such processes. This leads to a unified methodology to study the almost sure behavior of various statistics which are local functionals of the empirical distribution. Such statistics include density estimators and the Bahadur-Kiefer representation.