Log Log Laws for Empirical Measures

Abstract

Let $(X, \mathscr{A}, P)$ be a probability space and $\mathscr{C}$ a collection of measurable sets. Suppose $\mathscr{C}$ is a Donsker class, i.e., the central limit theorem for empirical measures holds uniformly on $\mathscr{C}$, in a suitable sense. Suppose also that suitable ($P\varepsilon$-Suslin) measurability conditions hold. Then we show that the $\log\log$ law for empirical measures, in the Strassen-Finkelstein form, holds uniformly on $\mathscr{C}$.