Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes
type: Item
instance of: scholarly article
title: Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes
zbMATH Open document ID: 0604.60032
author: Charles M. Goldie
author: Priscilla Greenwood
published in: The Annals of Probability
publication date: 1986-01-01T00:00:00Z
review text: Continuing their work ''Characterization of set-indexed Brownian motion and associated conditions for finite-dimensional convergence.'' ibid. 14, 802-816 (1986), the authors combine the results obtained there with a tightness lemma to give weak-convergence of continuous-path processes. To achieve this the index set is restricted, requiring that it satisfies a metric entropy condition [cf. \textit{R. M. Dudley}, ibid. 1, 66-103 (1973; Zbl 0261.60033)] in order that the processes dealt with and their Wiener limit can live in a space of continuous functions. The tightness proof requires hypotheses on the metric entropy of the index set, and \((2+\epsilon)\)-moments and a (severe) mixing rate for the summands. An extension to the dependent case of a method of \textit{R. F. Bass} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 591-608 (1985; Zbl 0575.60034)] is essentially used. As results of independent interest we note variance bounds on arbitrary sums of \(Z^ d\)-indexed mixing random variables, and uniform integrability conditions for such sums.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 3975998
zbMATH Keywords: tightness lemma
zbMATH Keywords: weak-convergence of continuous-path processes
zbMATH Keywords: mixing rate
zbMATH Keywords: uniform integrability conditions
reviewed by: Marius Iosifescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992440
OpenAlex ID: W2049207421
DOI: 10.1214/AOP/1176992440
description: scientific article; zbMATH DE number 3975998