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What is a multi-parameter renewal process? (English) Zbl 1109.60070

Summary: The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.

MSC:

60K05 Renewal theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G48 Generalizations of martingales
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