Palm theory for random time changes

Journal of Applied Mathematics and Stochastic Analysis. 2001;14(1):55-74 DOI 10.1155/S1048953301000065


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Journal Title: Journal of Applied Mathematics and Stochastic Analysis

ISSN: 1048-9533 (Print); 1687-2177 (Online)

Publisher: Hindawi Publishing Corporation

LCC Subject Category: Science: Mathematics



Masakiyo Miyazawa (Science University of Tokyo, Dept. of Information Sciences, Noda City, Chiba 278, Japan)
Gert Nieuwenhuis (Tilburg University, Dept. of Econometrics, PO Box 90153, LE Tilburg NL-5000, The Netherlands)
Karl Sigman (Columbia University, Dept. of Industrial Eng. and Operations Research, 500 West 120th Street, MC 4704, New York, NY 10027, USA)



Abstract | Full Text

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, we are able to characterize all of the above systems in a single framework. Moreover, this leads to what we 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.