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On the characterization of point processes with the order statistic property

Published online by Cambridge University Press:  14 July 2016

Paul D. Feigin
Paul D. Feigin*
Affiliation:
Technion — Israel Institute of Technology
*
∗Postal address: Faculty of Industrial and Management Engineering, Technion — Israel Institute of Technology, Haifa, Israel.
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Abstract

We provide a probabilistic proof of the characterization of point processes (on the real line) with the order statistic property. The characterization is used to investigate the homogeneity of such processes and is also related to the martingale theory associated with point processes.

Keywords

ORDER STATISTIC PROPERTYMIXED POISSON PROCESSESYULE PROCESSMARTINGALE THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 297 - 304
Copyright
Copyright © Applied Probability Trust 1979 

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References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Bremaud, P. and Jacod, J. (1977) Processus ponctuels et martingales: résultats récents sur la modélisation et le filtrage. Adv. Appl. Prob. 9, 382–416.CrossRefGoogle Scholar
Crump, K. (1975) On point processes having an order statistic property. Sankhya A 37, 396–404.Google Scholar
Epstein, B. (1962) Simple estimates of the parameters of exponential distributions, in Contributions to Order Statistics, ed. Sarhan, A. E. and Greenberg, B. C., Wiley, New York, 361–371.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications 2, 2nd edn. Wiley, New York.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Holmes, P. T. (1971) On a property of Poisson process. Sankhya A 33, 93–98.Google Scholar
Kallenberg, O. (1973) Characterization and convergence of random measures and point processes. Z. Warscheinlichkeitsth. 27, 9–21.Google Scholar
Keiding, N. (1974) Estimation in the birth process. Biometrika 61, 71–80.Google Scholar
Kendall, D. G. (1966) Branching processes since 1873. J. Lond. Math. Soc. 41, 385–406.Google Scholar
Nawrotzki, K. (1962) Eine Grenzwersatz für homogene zufällige Punktfolgen. Math. Nachr. 24, 201–217.Google Scholar
Neuts, M. and Resnick, S. I. (1971) On the times of birth in a linear birth process. J. Austral. Math. Soc. 12, 473–475.Google Scholar
Puri, P. S. (1968) Some further results on the birth-and-death process and its integral. Proc. Camb. Phil. Soc. 64, 141–154.CrossRefGoogle Scholar
Waugh, W. A. O'N. (1970) Transformation of a birth process into a Poisson process. J. R. Statist. Soc. B 32, 418–431.Google Scholar
Westcott, M. (1973) Some remarks on a property of the Poisson process. Sankhya A 35, 29–34.Google Scholar