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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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References

Chistyakov, A. P. (1964) A theorem on sums of independent positive random variables and its applications to branching processes. Theor. Prob. Appl. 9, 640–648.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973) Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.Google Scholar
Cline, D. B. H. (1987) Convolution of distributions with exponential and subexponential tails. J. Austral. Math. Soc. A43, 347–365.Google Scholar
Embrechts, P. and Goldie, C. M. (1980) On closure and factorisation properties of subexponential and related distributions. J. Austral. Math. Soc. A29, 243–256.Google Scholar
Feller, W. (1965) An Introduction to Probability Theory and its Applications , 2nd edn, Vol. 1, Wiley, New York.Google Scholar
Goldie, C. M. (1978) Subexponential distributions and dominated-variation tails. J. Appl. Prob. 15, 440–442.Google Scholar
Klüppelberg, C. (1988) Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Teugels, J. L. (1975) The class of subexponential distributions. Ann. Prob. 3, 1000–1011.Google Scholar