Journal of Applied Mathematics and Stochastic Analysis
Volume 14 (2001), Issue 1, Pages 55-74
Palm theory for random time changes
1Science University of Tokyo, Dept. of Information Sciences, Noda City, Chiba 278, Japan
2Tilburg University, Dept. of Econometrics, PO Box 90153, LE Tilburg NL-5000, The Netherlands
3Columbia University, Dept. of Industrial Eng. and Operations Research, 500 West 120th Street, MC 4704, New York 10027, NY, USA
Received 1 June 1998; Revised 1 June 2000
Copyright © 2001 Masakiyo Miyazawa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 in return.
To illustrate the generality of our approach, we show that classical
Palm theory for random measures is included in our RTC framework. We
also consider the important special case of marked point processes with
batches. We illustrate how our approach naturally allows one to distinguish between the marks within a batch while retaining nice stationarity