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Association and random measures. (English) Zbl 0681.60045
Our point of departure is the result, due to R. M. Burton and 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.
Reviewer: S.N.Evans

##### MSC:
 60G57 Random measures 60E15 Inequalities; stochastic orderings 60E07 Infinitely divisible distributions; stable distributions 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F05 Central limit and other weak theorems 60F15 Strong limit theorems
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