On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. (English) Zbl 1054.60012

Cai, Jun; Tang, Qihe

J. Appl. Probab. 41, No. 1, 117-130 (2004).

Heavy-tailed distributions can be used in various insurance and finance applications. The aim of the present paper is to investigate max-sum equivalence and convolution closure properties of certain classes of heavy-tailed distributions. The authors consider the following classes and denotations of heavy-tailed distributions: regular variation \((R)\), consistent variation \((C)\), dominant variation \((D)\), subexponential variation \((S)\), and long-tailed variation \((L)\) distributions. These classes are known to satisfy the following inclusion relations: \[ R\subset C\subset D\cap L\subset S\subset L. \] The main results of the paper consist in extending the properties of the max-sum equivalence and convolution to larger classes of heavy-tailed distributions, and applying these effects to the study of asymptotic behaviour of some important distributions. More precisely, in Section 2, the authors prove that the class \(D\) of heavy-tailed distributions is closed under convolution, and the max-sum equivalence property is shown to hold for the class \(D\cap L\). The class \(C\) is then shown to be closed under convolution. Section 3 considers the closure properties of \(C\) and \(D\cap L\) classes under compound distributions, in particular, under compound geometric distribution. Asymptotic behaviour of the tails of compound geometric convolutions is discussed. These results are applied in Section 4 to the ruin probability in the compound Poisson risk process by an \(\alpha\)-stable Lévy motion, and in Section 5 to the equilibrium waiting-time distribution of the M/G/\(k\) type queue.

Reviewer: Neculai Curteanu (Iaşi)

MSC:

60E05	Probability distributions: general theory
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
91B30	Risk theory, insurance (MSC2010)

Keywords:

max-sum equivalence; convolution closure; heavy-tailed distribution; consistent variation; subexponentiality; long-tailed distribution; compound geometric distribution; \(\alpha\)-stable Lévy process; Poisson risk process; ruin probability; equilibrium waiting-time; M/G/\(k\) queue

References:

[1]	Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. · Zbl 0617.26001
[2]	Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18, 1388–1402. JSTOR: · Zbl 0709.60016
[3]	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. Y. P. Chaubey, Imperial College Press, London, pp. 114–131. · Zbl 1270.60021
[4]	Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557. · Zbl 0577.60019
[5]	Cline, D. B. H. (1994). Intermediate regular and \(\Pi\) variation. Proc. Lond. Math. Soc. 68, 594–616. · Zbl 0793.26004
[6]	Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98. · Zbl 0799.60015
[7]	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. · Zbl 0723.62065
[8]	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. · Zbl 0425.60011
[9]	Embrechts, P. and Omey, E. (1984). A property of longtailed distributions, J. Appl. Prob. 21, 80–87. · Zbl 0534.60015
[10]	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. · Zbl 0518.62083
[11]	Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347. · Zbl 0397.60024
[12]	Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[13]	Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York. · Zbl 0219.60003
[14]	Furrer, H. (1998). Risk processes perturbed by \(\alpha\)-stable Lévy motion. Scand. Actuarial J. 1998, 59–74. · Zbl 1026.60516
[15]	Gertsbakh, I. B. (1984). Asymptotic methods in reliability theory: a review. Adv. Appl. Prob. 16, 147–175. · Zbl 0528.60085
[16]	Greiner, M., Jobmann, M. and Klüppelberg, C. (1999). Telecommunication traffic, queueing models, and subexponential distributions. Queueing Systems 33, 125–152. · Zbl 0997.60116
[17]	Jelenković, P. R. and Lazar, A. A. (1999). Asymptotic results for multiplexing subexponential on–off processes. Adv. Appl. Prob. 31, 394–421. · Zbl 0952.60098
[18]	Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer, Dordrecht. · Zbl 0881.60043
[19]	Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141. · Zbl 0651.60020
[20]	Kováts, A. and Móri, T. F. (1992). Ageing properties of certain dependent geometric sums. J. Appl. Prob. 29, 655–666. · Zbl 0755.62071
[21]	Leslie, J. (1989). On the non-closure under convolution of the subexponential family. J. Appl. Prob. 26, 58–66. · Zbl 0672.60027
[22]	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. · Zbl 0599.60084
[23]	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. · Zbl 1051.60032
[24]	Omey, E. (1994). On the difference between the product and the convolution product of distribution functions. Publ. Inst. Math. (Béograd) 55(69), 111–145. · Zbl 0824.60032
[25]	Schlegel, S. (1998). Ruin probabilities in perturbed risk models. Insurance Math. Econom. 22, 93–104. · Zbl 0907.90100
[26]	Schmidli, H. (2001). Distribution of the first ladder height of a stationary risk process perturbed by \(\alpha \)-stable Lévy motion. Insurance Math. Econom. 28, 13–20. · Zbl 0981.60041
[27]	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. · Zbl 1043.60012
[28]	Van Hoorn, M. H. (1984). Algorithms and Approximations for Queueing Systems (CWI Tract 8 ). Stichting Mathematisch Centrum, Amsterdam. · Zbl 0541.60095
[29]	Veraverbeke, N. (1993). Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance Math. Econom. 13, 57–62. · Zbl 0790.62098
[30]	Willmot, G. and Lin, X. (1996). Bounds on the tails of convolutions of compound distributions. Insurance Math. Econom. 18, 29–33. · Zbl 0853.62082
[31]	Yang, H. and Zhang, L. Z. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Prob. 33, 281–291. · Zbl 0978.60104