Association and random measures
type: Item
instance of: scholarly article
title: Association and random measures
zbMATH Open document ID: 0681.60045
DOI: 10.1007/BF01207510
author: Steven Neil Evans
publication date: 1990-01-01T00:00:00Z
review text: Our point of departure is the result, due to \textit{R. M. Burton} and \textit{E. Waymire} [Dependence in Probability and Statistics, Conf. Oberwolfach 1985, Prog. Probab. Stat. 11, 383-395 (1986; Zbl 0603.60041)] that every infinitely divisible random measure has the property variously known as association, positive correlations, or the FKG property. This leads to a study of stationary, associated random measures on \({\mathbb{R}}^ d\). We establish simple necessary and sufficient conditions for ergodicity and mixing when second moments are present. We also study the second moment condition that is usually referred to as finite susceptibility. As one consequence of these results, we can easily rederive some central limit theorems of Burton and Waymire. Using association techniques, we obtain a law of the iterated logarithm for infinitely divisible random measures under simple moment hypotheses. Finally, we apply these results to a class of stationary random measures related to measure-valued Markov branching processes.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4115650
zbMATH Keywords: infinite divisibility
zbMATH Keywords: association
zbMATH Keywords: FKG property
zbMATH Keywords: positive correlations
zbMATH Keywords: correlation inequality
zbMATH Keywords: mixing
zbMATH Keywords: ergodicity
zbMATH Keywords: central limit theorem
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: measure-valued branching process
zbMATH Keywords: superprocess
MaRDI profile type: Publication
cites work: On the convergence rate in the central limit theorem for associated processes
cites work: Scaling limits for associated random measures
cites work: Q3739970
cites work: A functional law of the iterated logarithm for associated sequences
cites work: The critical measure diffusion process
cites work: Q4194257
cites work: Association of Random Variables, with Applications
cites work: Q3490685
cites work: Q5624436
cites work: Construction and regularity of measure-valued Markov branching processes
cites work: Correlation inequalities on some partially ordered sets
cites work: A correlation inequality for Markov processes in partially ordered state spaces
cites work: Diffusion equation techniques in stochastic monotonicity and positive correlations
cites work: A weighted occupation time for a class of measure-valued branching processes
cites work: Q3333815
cites work: Association of probability measures on partially ordered spaces
cites work: Q4164078
cites work: Normal fluctuations and the FKG inequalities
cites work: An invariance principle for certain dependent sequences
cites work: Q4744504
cites work: Q4131340
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4115650