The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component

Abstract

Weak convergence results are obtained for empirical processes indexed by classes F \mathcal {F} of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes { f ( X ) : f ∈ F } \{ f(X):\,f \in \mathcal {F}\} ("bracketing" conditions). They extend (and slightly improve upon) a central limit theorem of Marcus and Pisier (1984) for Lipschitzian processes. The law of the iterated logarithm is also considered. The examples include Marcinkiewicz type laws of large numbers for weighted empirical processes and for the dual-bounded-Lipschitz distance between a probability in R {\mathbf {R}} and its associated empirical measures.