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On the non-closure under convolution of the subexponential family

Published online by Cambridge University Press:  14 July 2016

J. R. Leslie
J. R. Leslie*
Affiliation:
Birkbeck College
*
∗Postal address: Department of Statistics, Birkbeck College, University of London, Malet Street, London WC1E 7HX, UK.
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Abstract

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F(2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.

Keywords

CONVOLUTION CLOSURE
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 1 , March 1989 , pp. 58 - 66
Copyright
Copyright © Applied Probability Trust 1989 

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