Some Limit Theorems for Empirical Processes

Abstract

In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions F. From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions. One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition). The case of unbounded F is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case. The results for empiricals also yield some new CLT's in C[ 0, 1] and D[ 0, 1].