## The Annals of Probability

- Ann. Probab.
- Volume 14, Number 3 (1986), 817-839.

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

Charles M. Goldie and Priscilla E. Greenwood

#### 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].

#### Article information

**Source**

Ann. Probab., Volume 14, Number 3 (1986), 817-839.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992440

**Digital Object Identifier**

doi:10.1214/aop/1176992440

**Mathematical Reviews number (MathSciNet)**

MR841586

**Zentralblatt MATH identifier**

0604.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60E15: Inequalities; stochastic orderings 60B10: Convergence of probability measures

**Keywords**

Brownian motion lattice-indexed random variables metric entropy mixing random variables partial-sum processes set-indexed processes splitting $n$-dimensional sets tightness uniform integrability variance bounds weak convergence Wiener process

#### Citation

Goldie, Charles M.; Greenwood, Priscilla E. Variance of Set-Indexed Sums of Mixing Random Variables and Weak Convergence of Set-Indexed Processes. Ann. Probab. 14 (1986), no. 3, 817--839. doi:10.1214/aop/1176992440. https://projecteuclid.org/euclid.aop/1176992440