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The coincidence approach to stochastic point processes

Published online by Cambridge University Press:  01 July 2016

Odile Macchi
Odile Macchi*
Affiliation:
Laboratoire d'Etude des Phénomènes Aléatoires, Université de Paris-Sud
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Abstract

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Keywords

STOCHASTIC POINT PROCESSJOINT COUNTING EVENTCOINCIDENCE PROBABILITYCOINCIDENCE DENSITYEXCLUSION PROBABILITY DENSITYREGULAR PROCESSPHOTON PROCESSGAUSS-POISSON PROCESSFERMION PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 1 , March 1975 , pp. 83 - 122
Copyright
Copyright © Applied Probability Trust 1975 

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