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A characterization of the gamma distribution from a random difference equation

Published online by Cambridge University Press:  14 July 2016

Eric S. Tollar
Eric S. Tollar*
Affiliation:
The Florida State University
*
∗ Present address: Bell Communications Research, Rm 2Z-222, 331 Newman Springs Road, Red Bank, NJ 07701–7020, USA.
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Abstract

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

Keywords

INDEPENDENCE

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 142 - 149
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported in part by the U.S. Army Research Office Grant Number DAAG 29–82-K-0168.

The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.

References

Bernard, S. R., Shenton, L. R. and Uppuluri, V. R. R. (1967) Stochastic models for the distribution of radioactive material in a connected system of compartments. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 4, 481–510.Google Scholar
Cavalli-Sforza, L. and Feldman, M. W. (1973) Models for cultural inheritance I. Group mean and within group variation. Theoret. Popn Biol. 4, 42–55.Google Scholar
Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar
Lukacs, E. (1955) A characterization of the gamma distribution. Ann. Math. Statist. 26, 319–324.Google Scholar
Marsaglia, G. (1974) Extension and application of Lukacs' characterization of the gamma distribution. Symp. Stat. and Rel. Topics , ed. Selah, A. K., Carlton Univ., 9.01–9.13.Google Scholar
Paulson, A. S. and Uppuluri, V. R. R. (1972) Limit laws on a sequence determined by a random difference equation governing a one-compartment system. Math. Biosci. 13, 325–333.Google Scholar
Solomon, F. (1975) Random walks in a random environment. Ann. Prob. 3, 1–31.Google Scholar
Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar