Summary
Our point of departure is the result, due to Burton and Waymire, that every infinitely divisible random measure has the property variously known as association, positive correlations, or the FKG property. This leads into a study of stationary, associated random measures onR 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 referrent 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.
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Research supported in part by NSF Grant DMS-8701212 at the University of Virginia
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Evans, S.N. Association and random measures. Probab. Th. Rel. Fields 86, 1–19 (1990). https://doi.org/10.1007/BF01207510
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DOI: https://doi.org/10.1007/BF01207510