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Poisson mixtures and quasi-infinite divisibility of distributions

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri  and
Charles M. Goldie
Prem S. Puri*
Affiliation:
Purdue University
Charles M. Goldie*
Affiliation:
University of Sussex
*
∗Postal address: Department of Statistics, Purdue University, Mathematical Sciences Building, West Lafayette, IN 47907, U.S.A.
∗∗Postal address: Mathematics Division, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, U.K.
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Abstract

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.

Keywords

CUMULANTSINFINITE DIVISIBILITYMIXTURE OF DISTRIBUTIONSPOISSON MIXTUREQUASI-INFINITE DIVISIBILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 138 - 153
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

These investigations were started while this author was working on the project at the Statistics Laboratory, University of California, Berkeley, supported by the U.S. Energy Research and Development Agency and completed at Purdue University with the support of U.S. National Science Foundation Grant No. 0199–50–13995.

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