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Limits of compound and thinned point processes

Published online by Cambridge University Press:  14 July 2016

Olav Kallenberg
Olav Kallenberg*
Affiliation:
University of Göteborg
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Abstract

Let η =Σj δτj be a point process on some space S and let β,β 1,β 2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σj βj δτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

Keywords

COMPOUND AND THINNED POINT PROCESSESINFINITELY DIVISIBLE RANDOM MEASURESSUBORDINATED POISSON PROCESSESCONVERGENCE IN DISTRIBUTIONREGULARITY AND DIFFUSENESS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 269 - 278
Copyright
Copyright © Applied Probability Trust 1975 

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References

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