Donsker Classes and Random Geometry

Abstract

Let $\mathscr{F}$ be a class of square integrable functions. We give necessary and sufficient random geometric conditions for the empirical process indexed by $\mathscr{F}$ to satisfy the CLT. These conditions roughly mean that the trace of $\mathscr{F}$ on a random sample is a small (for the $l^1$ norm) perturbation of a set which is nice for the $l^2$ norm.