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Rarefactions of compound point processes

Published online by Cambridge University Press:  14 July 2016

Richard F. Serfozo
Richard F. Serfozo*
Affiliation:
AT & T Bell Laboratories
*
∗Present address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.
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Abstract

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

Keywords

POISSON PROCESSESCOX PROCESSESRANDOM MEASURESINFINITE DIVISIBILITYTHINNING OF POINT PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 4 , December 1984 , pp. 710 - 719
Copyright
Copyright © Applied Probability Trust 1984 

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References

Bauer, H. (1972) Probability Theory and Elements of Measure Theory. Holt, Rinehart, and Winston, New York.Google Scholar
Böker, F. and Serfozo, R. (1983) Ordered thinnings of point processes and random measures. Stoch. Proc. Appl. 15, 113–132.CrossRefGoogle Scholar
Brown, T. (1979) Position dependent and stochastic thinnings of point processes. Stoch. Proc. Appl. 9, 189–193.CrossRefGoogle Scholar
Jagers, P. and Lindvall, T. (1974) Thinning and rare events in point processes. Z. Wahrscheinlichkeitsth. 28, 89–98.CrossRefGoogle Scholar
Kallenberg, O. (1975a) Limits of compound and thinned processes. J. Appl. Prob. 12, 269–278.CrossRefGoogle Scholar
Kallenberg, O. (1975b) Random Measures. Akademie-Verlag, Berlin (and Academic Press, New York, 1976).Google Scholar
Renyi, A. (1956) A characterization of Poisson processes (in Hungarian). Magyar Tud. Akad. Mat. Kotato Int. Közl. 1, 519–527.Google Scholar
Serfozo, R. (1976) Compositions, inverses, and thinnings of random measures. Z. Wahrscheinlichkeitsth. 7, 253–265.Google Scholar