Characterisations of set-indexed Brownian motion and associated conditions for finite-dimensional convergence
type: Item
instance of: scholarly article
title: Characterisations of set-indexed Brownian motion and associated conditions for finite-dimensional convergence
zbMATH Open document ID: 0603.60017
author: Charles M. Goldie
author: Priscilla Greenwood
published in: The Annals of Probability
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathcal A}\) be a subset of the \(\sigma\)-algebra of Borel subsets of \([0,1]^ d\). A process X on \({\mathcal A}\) is a collection \(\{X(A)\}_{A\in {\mathcal A}}\) of real-valued random variables. A standard Wiener process, or Brownian motion, on \({\mathcal A}\) is a Gaussian process W with the properties \(EW(A)=0\), \(cov(W(A),W(B))=| A\cap B|\) for A,B\(\in {\mathcal A}\), where \(| \cdot |\) is Lebesgue measure. The main results of the authors are characterizations of W on \({\mathcal R} = the\) class of all finite unions of left-open right-closed intervals in \([0,1]^ d\), and weak convergence to it. From these characterization and convergence results relative to an arbitrary class \({\mathcal A}\) are deduced. The results obtained are used to give new multidimensional central limit theorems for mixing random fields. They need only second- moment assumptions on the summands, a logarithmic mixing rate, and convergence of second moments of certain special partial sums.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 62E10
zbMATH DE Number: 3973946
zbMATH Keywords: weak convergence
zbMATH Keywords: logarithmic mixing rate
reviewed by: Marius Iosifescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992439
OpenAlex ID: W2092500187
DOI: 10.1214/AOP/1176992439
description: scientific article; zbMATH DE number 3973946