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On some characterizations of the poisson process

Published online by Cambridge University Press:  14 July 2016

Wen-Jang Huang
Wen-Jang Huang*
Affiliation:
National Sun Yat-Sen University
*
∗Postal address: Institute of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, ROC.
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Abstract

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

Keywords

THINNED POINT PROCESSRENEWAL PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 1 , March 1989 , pp. 176 - 181
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

This research was supported by the National Science Council under the Grant NSC-76-0208-M110-03.

References

Chandramohan, J. and Liang, L. K. (1985) Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes. J. Appl. Prob. 22, 828–835.Google Scholar
Chandramohan, J., Foley, R. D. and Disney, R. L. (1985) Thinning of point processes — covariance analysis. Adv. Appl. Prob. 17, 127–146.Google Scholar
Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
Kimeldorf, G. and Thall, P. F. (1983) A joint characterization of the multinomial distribution and the Poisson process. J. Appl. Prob. 20, 202–208.Google Scholar