Count distributions, orderliness and invariance of Poisson cluster processes
- 1 June 1979
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 16 (2) , 261-273
- https://doi.org/10.2307/3212895
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
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