Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T11:50:35.119Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

Albrecht, P. (1985) Mixed Poisson processes. In Encyclopedia of Statistical Sciences, Vol. 6, Wiley, New York.Google Scholar
Bernstein, S. (1928) Sur les fonctions absolument monotones. Acta Math. 51, 1–66.Google Scholar
Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
Lundberg, O. (1940) On Random Processes and Their Application to Sickness and Accident Statistics. Reprinted (1964) by Almqvist and Wiksell, Uppsala.Google Scholar
Pfeifer, D. (1982) The structure of elementary pure birth processes. J. Appl. Prob. 19, 664–667.Google Scholar
Pfeifer, D. (1986) Polya–Lundberg process. Encyclopedia of Statistical Sciences, Vol. 7, Wiley, New York.Google Scholar