Comparison Theorems, Random Geometry and Some Limit Theorems for Empirical Processes

Abstract

In this paper, we obtain several new results and developments in the study of empirical processes. A comparison theorem for Rademacher averages is at the basis of the first part of the results, with applications, in particular, to Kolmogorov's law of the iterated logarithm and Prokhorov's law of large numbers for empirical processes. We then study the behavior of empirical processes along a class of functions through random geometric conditions and complete in this way the characterization of the law of the iterated logarithm. Bracketing and local Lipschitz conditions provide illustrations of some of these ideas to concrete situations.