Limit Theorems for Weakly Dependent Random Vectors

Abstract

In Chapter I we improve upon results on the almost sure approximation of the empirical process of weakly dependent random vectors, recently obtained by Berkes and Philipp and Philipp and Pinzur. For strongly mixing sequences we relax the bounds on the mixing rates, and for absolutely regular sequences we improve the error term. We also extend these results to random vectors which are functions of the given sequence as well as to random variables which are evaluated at lacunary sequences.

In Chapter 2 we extend the uniform law of the iterated logarithm for classes of functions in Lip (alpha) ((alpha) > 1/2) evaluated at lacunary sequences and functions of mixing processes, due to Kaufman and Philipp. Here we relax the lacunarity condition and the mixing rates.