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

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.

[1]  Asymptotic estimates for the probability of ruin in a Poisson model with diffusion , 1993 .

[2]  P. Embrechts,et al.  On closure and factorization properties of subexponential and related distributions , 1980, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[3]  Qihe Tang,et al.  Characterizations on Heavy-tailed Distributions by Means of Hazard Rate , 2003 .

[4]  PAUL EMBRECHTS,et al.  Modelling of extremal events in insurance and finance , 1994, Math. Methods Oper. Res..

[5]  Mark Brown.  Error bounds for exponential approximations of geometric convolutions , 1990 .

[6]  ChunSu,et al.  Characterizations on Heavy—tailed Distributions by Means of Hazard Rate , 2003 .

[7]  Sabine Schlegel.  Ruin probabilities in perturbed risk models , 1998 .

[8]  M. Miyazawa.  Approximation of the queue-length distribution of an M/GI/s queue by the basic equations , 1986, Journal of Applied Probability.

[9]  Claudia Klüppelberg,et al.  Telecommunication traffic, queueing models, and subexponential distributions , 1999, Queueing Syst. Theory Appl..

[10]  Gennady Samorodnitsky,et al.  Subexponentiality of the product of independent random variables , 1994 .

[11]  Hansjörg Furrer,et al.  Risk processes perturbed by α-stable Lévy motion , 1998 .

[12]  J. Doob.  Stochastic processes , 1953 .

[13]  Paul Embrechts,et al.  A PROPERTY OF LONGTAILED DISTRIBUTIONS , 1984 .

[14]  I. Gertsbakh.  Asymptotic methods in reliability theory: a review , 1984, Advances in Applied Probability.

[15]  Bounds on the tails of convolutions of compound distributions , 1996 .

[16]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[17]  Hailiang Yang,et al.  Spectrally negative Lévy processes with applications in risk theory , 2001, Advances in Applied Probability.

[18]  Charles M. Goldie,et al.  Subexponentiality and infinite divisibility , 1979 .

[19]  C. Klüppelberg.  Subexponential distributions and integrated tails. , 1988 .

[20]  W. A. Thompson,et al.  Algorithms and Approximations for Queueing Systems. , 1985 .

[21]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[22]  P. Embrechts,et al.  Estimates for the probability of ruin with special emphasis on the possibility of large claims , 1982 .

[23]  T. F. Móri,et al.  Ageing properties of certain dependent geometric sums , 1992, Journal of Applied Probability.

[24]  A. Lazar,et al.  Asymptotic results for multiplexing subexponential on-off processes , 1999, Advances in Applied Probability.

[25]  D. B. Cline,et al.  Intermediate Regular and Π Variation , 1994 .

[26]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[27]  V. Kalashnikov,et al.  Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing , 1997 .

[28]  Daren B. H. Cline,et al.  Convolution tails, product tails and domains of attraction , 1986 .

[29]  Hanspeter Schmidli.  Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion , 2001 .

[30]  J. Leslie,et al.  On the non-closure under convolution of the subexponential family , 1989, Journal of Applied Probability.

[31]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[32]  Hailiang Yang,et al.  Precise large deviations for sums of random variables with consistently varying tails , 2004, Journal of Applied Probability.

[33]  J. Geluk.  Π-regular variation , 1981 .