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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, X 1, X 2, … are non-constant mutually independent random variables with X 1,X 2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X 1 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

Information

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
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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
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