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On Cox processes and gamma renewal processes

Published online by Cambridge University Press:  14 July 2016

Nikos Yannaros
Nikos Yannaros*
Affiliation:
Royal Institute of Technology, Stockholm
*
∗Postal address: Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden.
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Abstract

It is shown that the gamma distribution with shape parameter α can be obtained through a p-thinning for every 0 < p < 1, when 0 < α ≦ 1. In the case α > 1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0 < α ≦ 1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.

Keywords

POINT PROCESSESTHINNINGGAMMA DISTRIBUTIONINVERSES OF THINNED PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 25 , Issue 2 , June 1988 , pp. 423 - 427
Copyright
Copyright © Applied Probability Trust 1988 

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References

Berg, C. and Forst, G. (1975) Potential Theory on Locally Compact Abelian Groups. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Bochner, S. (1955) Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. II, 2nd edn. Wiley, New York.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Springer-Verlag, Berlin.Google Scholar
Kingman, J. F. C. (1964) On doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.CrossRefGoogle Scholar
Kolsrud, T. (1985) Some comments on thinned renewal processes. Dept of Mathematics, Univ. of Stockholm.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York.Google Scholar
Thedeen, T. (1986) The inverses of thinned point processes. Dept. of Statistics, University of Stockholm, Research Report 1986:1.Google Scholar
Yannaros, N. (1985) On the thinning of renewal point processes. Dept. of Statistics, University of Stockholm, Research Report 1985:6.Google Scholar
Yannaros, N. (1986a) On Cox processes and renewal processes. Dept. of Statistics, University of Stockholm, Research Report 1986:3.Google Scholar
Yannaros, N. (1986b) The inverses of thinned renewal processes. Dept. of Statistics, University of Stockholm, Research Report 1986:8.Google Scholar