The Annals of Probability

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

Charles M. Goldie and Priscilla E. Greenwood

Full-text: Open access

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


Export citation