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Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes

Published online by Cambridge University Press:  14 July 2016

J. Chandramohan  and
Lung-Kuang Liang
J. Chandramohan
Affiliation:
Case Western Reserve University
Lung-Kuang Liang*
Affiliation:
Case Western Reserve University
*
∗∗ Postal address: Department of Operations Research, Case Western Reserve University, Cleveland, OH 44106, USA.
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Abstract

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

Keywords

DOUBLY STOCHASTIC POISSON PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 828 - 835
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

∗

Present address: WB 3B-124, AT&T Bell Laboratories, Holmdel, NJ 07733, USA.

References

Chandramohan, J., Foley, R. D. and Disney, R. L. (1985) Thinning of point processes-covariance analyses. Adv. Appl. Prob. 17, 127–146.Google Scholar
Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. 2 Wiley New York.Google Scholar