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Palm theory for random time changes. (English) Zbl 0984.60063

From the authors’ abstract: Palm distributions are basic tools when studying stationarity in the context of point processes, queueing systems, fluid queues or random measures. The framework varies with the random phenomenon of interest, but usually a one-dimensional group of measure-preserving shifts is the starting point. In the present paper, by alternatively using a framework involving random time changes (RTCs) and a two-dimensional family of shifts, the authors are able to characterize all of the above systems in a single framework. Moreover, this leads to what they call the detailed Palm distribution (DPD) which is stationary with respect to a certain group of shifts. The DPD has a very natural interpretation as the distribution seen at a randomly chosen position on the extended graph of the RTC, and satisfies a general duality criterion: the DPD of the DPD gives the underlying probability \(P\) in return. To illustrate the generality of their approach, the authors show that classical Palm theory for random measures is included in their RTC framework. They also consider the important special case of marked point processes with batches.
Reviewer: V.Schmidt (Ulm)

MSC:

60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G10 Stationary stochastic processes
60K25 Queueing theory (aspects of probability theory)
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