Scaling limits for associated random measures
type: Item
instance of: scholarly article
title: Scaling limits for associated random measures
zbMATH Open document ID: 0579.60039
published in: The Annals of Probability
publication date: 1985-01-01T00:00:00Z
review text: A random measure \(X=\{X(B):\) \(B\in {\mathcal B}({\mathbb{R}}^ n)\}\) is associated if for bounded Borel sets \(B_ 1,...,B_ k\), the random variables \(X(B_ 1),...,X(B_ k)\) are associated in the sense that for functions f and g, both increasing with respect to the coordinatewise partial ordering on \({\mathbb{R}}^ n\), \(Cov(f(X(B_ 1),..., X(B_ k)), g(X(B_ 1),..., X(B_ k)))\geq 0.\) This is shown to be equivalent to Cov(F(X),G(X))\(\geq 0\) for all functionals F,G that are increasing in the sense that \(F(\mu)\leq F(\nu)\) whenever \(\mu(A)\leq \nu (A)\) for each set A (and analogously for G). The authors' main result is a central limit theorem for stationary, associated random measures: for such a random measure X, if in addition \(E[X(B)^ 2]<\infty\) for each bounded set B, and (with \(I=[0,1]^ n)\) \(\sum_{k\in {\mathbb{Z}}^ d}Cov(X(I),X(I+k))=\eta <\infty\), then for disjoint rectangles \(A_ 1,...,A_ k\), the random vectors \(\lambda^{- n/2}(\{X(A_ 1) - E[X(A_ 1)]\},..., \{X(A_ k) - E[X(A_ k)]\})\) converge as \(\lambda\to \infty\) to a multivariate normal distribution with mean zero and covariance matrix \(\eta diag(| A_ 1|,...,| A_ k|).\) The proof employs a reduction to a limit theorem of \textit{C. M. Newman} [Commun. Math. Phys. 74, 119-128 (1980; Zbl 0429.60096)]. Applications to Poisson center cluster random measures, critical branching point processes, dependent thinning, and doubly stochastic point processes are presented.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 3928008
zbMATH Keywords: random measure
zbMATH Keywords: central limit theorem
zbMATH Keywords: Poisson center cluster random measures
zbMATH Keywords: branching point processes
zbMATH Keywords: dependent thinning
zbMATH Keywords: doubly stochastic point processes
author: Robert Burton
author: Edward C. Waymire
reviewed by: Alan F. Karr
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992810
OpenAlex ID: W1967754295
DOI: 10.1214/AOP/1176992810
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description: scientific article; zbMATH DE number 3928008