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Recent results on characterization of probability distributions: a unified approach through extensions of Deny&s theorem

Published online by Cambridge University Press:  01 July 2016

C. Radhakrishna Rao  and
D. N. Shanbhag
C. Radhakrishna Rao*
Affiliation:
University of Pittsburgh
D. N. Shanbhag*
Affiliation:
University of Sheffield
*
∗Postal address: Center for Multivariate Analysis, Fifth Floor, Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260, USA.
∗∗Postal address: Department of Probability and Statistics, University of Sheffield, Sheffield S3 7RH, UK.
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Abstract

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

Keywords

EXCHANGEABLE RANDOM VARIABLESEXPONENTIAL, GEOMETRIC, PARETO AND STABLE DISTRIBUTIONSINTEGRATED CAUCHY FUNCTIONAL EQUATIONSLAU–RAO THEOREMSHANBHAG&S LEMMA
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 660 - 678
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

This work was supported by the Air Force Office of Scientific Research under Contract F49620-82-K-0001. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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