Functional central limit theorems for triangular arrays of function-indexed processes under uniformly integrable entropy conditions

[摘要] Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an 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 processes will be also regained. Applications concern, in particular, so-called random measure processes which include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes. (C) 1997 Academic Press.

[发布日期] 1997-08-01

[关键词] functional central limit theorem;asymptotic equicontinuity;symmetrization;metric entropy;VC graph class;maximal inequality;empirical processes;partial sum processes;random measure processes;sequential empirical process;smoothing by convolution;nonparametric regression;intensity estimation