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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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References

Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18, 1388–1402.CrossRefGoogle Scholar
Cai, J., and Garrido, J. (2002). Asymptotic forms and bounds for tails of convolutions of compound geometric distributions, with applications. In Recent Advances in Statistical Methods, ed. Chaubey, Y. P., Imperial College Press, London, pp. 114–131.CrossRefGoogle Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.CrossRefGoogle Scholar
Cline, D. B. H. (1994). Intermediate regular and Π variation. Proc. Lond. Math. Soc. 68, 594–616.CrossRefGoogle Scholar
Cline, D. B. H., and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98.CrossRefGoogle Scholar
Dufresne, F., and Gerber, H. U. (1991). Risk theory for the compound Poisson process that is perturbed by a diffusion. Insurance Math. Econom. 10, 51–59.CrossRefGoogle Scholar
Embrechts, P., and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. A 29, 243–256.CrossRefGoogle Scholar
Embrechts, P., and Omey, E. (1984). A property of longtailed distributions, J. Appl. Prob. 21, 80–87.CrossRefGoogle Scholar
Embrechts, P., and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M., and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1998, 59–74.CrossRefGoogle Scholar
Gertsbakh, I. B. (1984). Asymptotic methods in reliability theory: a review. Adv. Appl. Prob. 16, 147–175.CrossRefGoogle Scholar
Greiner, M., Jobmann, M. and Klüppelberg, C. (1999). Telecommunication traffic, queueing models, and subexponential distributions. Queueing Systems 33, 125–152.CrossRefGoogle Scholar
Jelenković, P. R., and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on–off processes. Adv. Appl. Prob. 31, 394–421.CrossRefGoogle Scholar
Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer, Dordrecht.CrossRefGoogle Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.CrossRefGoogle Scholar
Kováts, A. and Móri, T. F. (1992). Ageing properties of certain dependent geometric sums. J. Appl. Prob. 29, 655–666.CrossRefGoogle Scholar
Leslie, J. (1989). On the non-closure under convolution of the subexponential family. J. Appl. Prob. 26, 58–66.CrossRefGoogle Scholar
Miyazawa, M. (1986). Approximation of the queue-length distribution of an M/GI/s queue by the basic equations. J. Appl. Prob. 23, 443–458.CrossRefGoogle Scholar
Ng, K., Tang, Q., Yan, J., and Yang, H. (2004). Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Prob. 41, 93–107.CrossRefGoogle Scholar
Omey, E. (1994). On the difference between the product and the convolution product of distribution functions. Publ. Inst. Math. (Béograd) 55(69), 111–145.Google Scholar
Schlegel, S. (1998). Ruin probabilities in perturbed risk models. Insurance Math. Econom. 22, 93–104.CrossRefGoogle Scholar
Schmidli, H. (2001). Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion. Insurance Math. Econom. 28, 13–20.CrossRefGoogle Scholar
Su, C., and Tang, Q. (2003). Characterizations on heavy-tailed distributions by means of hazard rate. Acta Math. Appl. Sinica English Ser. 19, 135–142.CrossRefGoogle Scholar
Van Hoorn, M. H. (1984). Algorithms and Approximations for Queueing Systems (CWI Tract 8). Stichting Mathematisch Centrum, Amsterdam.Google Scholar
Veraverbeke, N. (1993). Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance Math. Econom. 13, 57–62.CrossRefGoogle Scholar
Willmot, G., and Lin, X. (1996). Bounds on the tails of convolutions of compound distributions. Insurance Math. Econom. 18, 29–33.CrossRefGoogle Scholar
Yang, H., and Zhang, L. Z. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Prob. 33, 281–291.CrossRefGoogle Scholar