Variance of Set-Indexed Sums of Mixing Random Variables and Weak Convergence of Set-Indexed Processes

Abstract

A uniform bound is found for the variance of a partial-sum set-indexed process under a mixing condition. Sufficient conditions are given for a sequence of partial-sum set-indexed processes to converge to Brownian motion. The requisite tightness follows from hypotheses on the metric entropy of the class of sets and moment and mixing conditions on the summands. The proof uses a construction of Bass [2]. Convergence of finite-dimensional laws in this context is studied in [16].