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A martingale characterization of mixed Poisson processes

Published online by Cambridge University Press:  14 July 2016

Dietmar Pfeifer  and
Ursula Heller
Dietmar Pfeifer*
Affiliation:
Technical University Aachen
Ursula Heller
Affiliation:
Technical University Aachen
*
∗Postal address: Institut für Statistik und Wirtschaftsmathematik, RWTH Aachen, Wüllnerstrase 3, D-5100 Aachen, W. Germany.
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Abstract

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

Keywords

INTENSITIESCOMPLETELY MONOTONIC FUNCTIONSBERNSTEIN'S THEOREM
Type
Short Communications
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 246 - 251
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

Research supported in part by US AFOSR Contract No. F 49620 85 C 0144 at the University of North Carolina at Chapel Hill.

∗∗

Present address: Gothaer Lebensversicherung a.G., Gothaer Platz, 3400 Göttingen, W. Germany.

References

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Pfeifer, D. (1986) Polya–Lundberg process. Encyclopedia of Statistical Sciences, Vol. 7, Wiley, New York.Google Scholar