Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T01:50:21.916Z Has data issue: false hasContentIssue false
  • English
  • Français

A martingale characterization of Pólya-Lundberg processes

Part of: Stochastic processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Birgit Niese
Birgit Niese*
Affiliation:
Darmstadt University of Technology
*
∗Postal address: Department of Mathematics, Darmstadt University of Technology, Schloßgartenstraße 7, D-64289 Darmstadt, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

Keywords

Mixed Poisson processPólya-Lundberg processexponential family

MSC classification

Primary: 60G55: Point processes
Secondary: 62E99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 741 - 754
Copyright
© Applied Probability Trust 2006 

References

Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Grandell, J. (1997). Mixed Poisson Processes. Chapman and Hall, London.CrossRefGoogle Scholar
Grigelionis, B. (1998). On mixed Poisson processes and martingales. Scand. Actuarial J. 1998, 81–88.CrossRefGoogle Scholar
Hayakawa, Y. (2000). A new characterization property of mixed Poisson processes via Berman's theorem. J. Appl. Prob. 37, 261–268.CrossRefGoogle Scholar
Küchler, I. and Küchler, U. (1981). An analytical treatment of exponential families of stochastic processes with independent stationary increments. Math. Nachr. 102, 21–30.CrossRefGoogle Scholar
Küchler, U. and Sørensen, M. (1997). Exponential Families of Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Liptser, R. S. and Shiryayev, A. N. (1978). Statistics of Random Processes. II. Applications. Springer, New York.Google Scholar
Nawrotzki, K. (1962). Ein Grenzwertsatz für homogene zufällige Punktfolgen. Math. Nachr. 24, 201–217.Google Scholar
Pfeifer, D. and Heller, U. (1987). A martingale characterization of mixed Poisson processes. J. Appl. Prob. 24, 246–251.Google Scholar
Ycart, B. (1992). Integer valued Markov processes and exponential families. Statist. Prob. Lett. 14, 71–78.Google Scholar