Empirical Process Methods in Probability in Banach Spaces

Abstract

The purpose of this chapter is to present applications of the random process techniques developed so far to infinite dimensional limit theorems, and in particular to the central limit theorem (CLT). More precisely, we will be interested for example in the CLT in the space C(T) of continuous functions on a compact metric space T. Since C(T) is not well behaved with respect to the type or cotype 2 properties, we will rather have to seek for nice classes of random variables in C(T) for which a central limit property can be established. This point of view leads to enlarge this framework and to investigate limit theorems for empirical measures or processes. Random geometric descriptions of the CLT may then be produced via this approach, as well as complete descriptions for nice classes of functions (indicator functions of some sets) on which the empirical processes are indexed. While these random geometric descriptions do not solve the central limit problem in infinite dimension (and are probably of little use in applications), however, they clearly describe the main difficulties inherent to the problem from the empirical point of view.