FUNCTIONAL CENTRAL LIMIT-THEOREMS FOR TRIANGULAR ARRAYS OF FUNCTION-INDEXED PROCESSES UNDER UNIFORMLY INTEGRABLE ENTROPY CONDITIONS

Functional central limit theorems for triangular arrays of rowwise ind ependent stochastic processes are established by a method replacing ta il probabilities by expectations throughout. The main tool is a maxima l inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by a n additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-called, measure-like pr ocesses will be also regained. Applications concern, in particular, so -called random measure processes which include function-indexed empiri cal processes and partial-sum processes (with random or fixed location s). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examp les include nonparametric regression and intensity estimation for spat ial Poisson processes. (C) 1997 Academic Press.