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A joint characterization of the multinomial distribution and the Poisson process

Published online by Cambridge University Press:  14 July 2016

George Kimeldorf  and
Peter F. Thall
George Kimeldorf*
Affiliation:
The University of Texas at Dallas
Peter F. Thall*
Affiliation:
George Washington University
*
∗ Postal address: Programs in Mathematical Sciences, University of Texas at Dallas, Box 688, Richardson, TX 75080, U.S.A.
∗∗ Postal address: Department of Statistics, Bldg. C, Room 304, George Washington University, Washington, DC 20052, U.S.A.
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Abstract

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

Keywords

POINT PROCESSMULTINOMIAL DISTRIBUTIONTHINNED POINT PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 202 - 208
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research partly supported by NSF Grant MCS-8102564.

References

Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Fichtner, K. H. (1975) Charakterisierung Poissonscher zufälliger Punktfolgen und infinitesimale Verdünnungsschemata. Math. Nachr. 68, 93–104.CrossRefGoogle Scholar
Kallenberg, O. (1976) Random Measures. Akademie-Verlag, Berlin; Academic Press, London.Google Scholar
Kimeldorf, G., Plachky, D. and Sampson, A. (1981) A simultaneous characterization of the Poisson and Bernoulli distributions. J. Appl. Prob. 18, 316–320.CrossRefGoogle Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York; Akademie-Verlag, Berlin.Google Scholar