Limit Theorems for Empirical Processes Based on Dependent Data

Abstract

Empirical processes for non ergodic data are investigated under uniform distance. Some CLTs, both uniform and non uniform, are proved. In particular, conditions for Bn = n^(1/2) (µn - bn) and Cn = n^(1/2) (µn - an) to converge in distribution are given, where µn is the empirical measure, an the predictive measure, and bn = 1/n sum (ai) for i=0 to n-1. Such conditions can be applied to any adapted sequence of random variables. Various examples and a characterization of conditionally identically distributed sequences are given as well.Conditional identity in distribution, empirical process, exchangeability, predictive measure, stable convergence.