Large Deviations, Moderate Deviations and LIL for Empirical Processes

Abstract

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. r.v.'s with values in a measurable space $(E, \mathscr{E})$ of law $\mu$, and consider the empirical process $L_n(f) = (1/n)\sum^n_{k=1} f(X_k)$ with $f$ varying in a class of bounded functions $\mathscr{F}$. Using a recent isoperimetric inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of $L_n(\cdot)$ in the Banach space of bounded functionals $\mathscr{l}_\infty(\mathscr{F})$. The extension to the unbounded functionals is also discussed.