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Count distributions, orderliness and invariance of Poisson cluster processes

Published online by Cambridge University Press:  14 July 2016

Larry P. Ammann  and
Peter F. Thall
Larry P. Ammann*
Affiliation:
A. D
Peter F. Thall*
Affiliation:
University of Texas at Dallas
*
∗Postal address: Programs in Mathematical Sciences, University of Texas at Dallas, Box 688, Richardson, TX 75080, U.S.A.
∗Postal address: Programs in Mathematical Sciences, University of Texas at Dallas, Box 688, Richardson, TX 75080, U.S.A.
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Abstract

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Keywords

POINT PROCESSPOISSON CLUSTER PROCESSPROBABILITY GENERATING FUNCTIONALORDERLINESSFINITE-DIMENSIONAL DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 261 - 273
Copyright
Copyright © Applied Probability Trust 1979 

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