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On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications

Part of: Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Jun Cai  and
Qihe Tang
Jun Cai*
Affiliation:
University of Waterloo
Qihe Tang*
Affiliation:
University of Amsterdam
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email address: jcai@math.uwaterloo.ca
∗∗ Postal address: Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. Email address: q.tang@uva.nl
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Abstract

In this paper, we discuss max-sum equivalence and convolution closure of heavy-tailed distributions. We generalize the well-known max-sum equivalence and convolution closure in the class of regular variation to two larger classes of heavy-tailed distributions. As applications of these results, we study asymptotic behaviour of the tails of compound geometric convolutions, the ruin probability in the compound Poisson risk process perturbed by an α-stable Lévy motion, and the equilibrium waiting-time distribution of the M/G/k queue.

Keywords

Max-sum equivalenceconvolution closureheavy-tailed distributionconsistent variationsubexponentialitycompound geometric distributionα-stable Lévy processruin probabilityM/G/k queue

MSC classification

Secondary: 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 117 - 130
Copyright
Copyright © Applied Probability Trust 2004 

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