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Thinning of Point Processes—Martingale Method

Published online by Cambridge University Press:  27 July 2009

Shengwu He  and
Jiagang Wang
Shengwu He
Affiliation:
Partment of Mathematical Statistics East China Normal University Shanghai, China & Department of Mathematics Purdue University West Lafayette, Indiana 47907
Jiagang Wang
Affiliation:
Institute of Mathematics Fundan University Shanghai, China
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By using the martingale method, we show that thinning of an arbitrary point process produces independent thinned processes if and only if the original point process is (nonhomogeneous) Poisson. Thinning is a classical problem for point processes. It is well-known that independent homogeneous Poisson processes result from constant Bernoulli thinnings of homogeneous Poisson processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. It is easy to see that the converse is also true, i.e., if the thinned processes are independent nonhomogeneous Poisson processes, so are the original processes. But if we only suppose the thinned processes are independent, nothing is concerned with their distribution law, the problem of whether or not the original processes are (nonhomogeneous) Poisson becomes interesting and challenging, which is the objective of this paper. It is considerably surprising for us to arrive at the affirmative answer. So far as this problem is concerned, the most work was done under the renewal assumption. For example, Bremaud [1] showed that for arbitrary delayed renewal processes, the existence of a pair of uncorrelated thinned processes is sufficient to guarantee that the original process is Poisson. It is natural that the mathematical tools to solve the problem in this case be typical ones for renewal theory, such as renewal equations and Laplace-Stieltjes transformations. Obviously, they are not available for nonrenewal processes. We find out that the martingale method is the most efficient one to solve this problem in the general case. More precisely, we use mainly the dual predictable projections of point processes. In fact, the distribution law of a point process is determined uniquely by its dual predictable projection (see [5]), and Ma [6] offered us a very useful criterion of independence of jump processes having no common jump time through their dual predictable projections. Based on these results, it is not a long way to arrive at the destination.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 4 , Issue 1 , January 1990 , pp. 117 - 129
Copyright
Copyright © Cambridge University Press 1990

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References

Bremaud, P. (1975). An extension of Watanabe's characterization theorem. Journal of Applied Probability 8: 396–399.CrossRefGoogle Scholar
Chandramohan, J. & Liang, L.K. (1985). Bernoulli, multinomial, and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes. Journal of Applied Probability 22: 828–835.CrossRefGoogle Scholar
He, S.W. & Wang, J.G. (1982). The property of predictable representation of the sum of independent semimartingales. Zeilschrift für Wahrscheinlickeitscheorie und Verwandte Gebiete 61: 141–152.Google Scholar
He, S.W. & Wang, J.G. (1984). Two results on jump processes. Séminaire de Probabilités XVIII; Lecture Notes in Mathematics 1059: 256–267.Google Scholar
Jacod, J. (1975). Multivariate point processes: predictable projection, Radon-Nikodym derivative, representation of martingales. Zeitschrift für Wahrscheinlickeitscheorie und Verwandte Gebiete 31: 235–253.CrossRefGoogle Scholar
Ma, Z.M. (1986). Queues and marked point processes. Chinese Journal of Applied Probability and Statistics 2: 52–58.Google Scholar
Savits, T.H. (1988). Some multivariate distributions derived from a nonfatal shock model. Journal of Applied Probability 25: 383–390.CrossRefGoogle Scholar
Steele, J.M. (1973). When successes and failures are independent, a compound process is Poisson. The American Statistician 27: 232.CrossRefGoogle Scholar
Wang, J.G. (1981). Some remarks on processes with independent increments. Séminaire de Probabilités XV: Lecture Notes in Mathematics 850: 627–631.Google Scholar
Watanabe, S. (1964). Additive functionals of Markov processes and Levy systems. Japan Journal of Mathematics 34: 53–79.Google Scholar