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Scaling limits for associated random measures. (English) Zbl 0579.60039
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 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.
Reviewer: A.Karr

##### MSC:
 60G57 Random measures 60F05 Central limit and other weak theorems 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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