Limit theorems for a class of processes generalizing the U-empirical process

Abstract

In this paper, we develop theory and tools for studying U-processes, a natural higher-order generalization of the empirical processes. We introduce a class of random discrete U-measures that generalize the empirical U-measure. We establish a Glivenko-Cantelli and a Donsker theorem under conditions on entropy numbers prevalent in the theory of empirical processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The uniform limit theorems discussed in this paper are key tools for many further developments involving empirical process techniques. Our results are applied to prove the asymptotic normality of Liu's simplicial median. We conclude this paper by extending Anscombe's central limit theorem to encompass randomly stopped U-processes, building upon its application to randomly stopped sums of independent random variables.