Scaling limits for associated random measures
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.
scientific article; zbMATH DE number 3928008