### Log Log Laws for Empirical Measures

J. Kuelbs and R. M. Dudley
Source: Ann. Probab. Volume 8, Number 3 (1980), 405-418.

#### 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 $\mathscr{C}$.

First Page:
Primary Subjects: 60F15
Secondary Subjects: 60F05, 28A05
Full-text: Open access