A wide class of heavy-tailed distributions and its applications

Abstract

Let F(x) be a distribution function supported on [0, ∞) with an equilibrium distribution function F_e(x). In this paper we pay special attention to the hazard rate function r_e(x) of F_e(x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of r_e(x) we give a criterion to identify F(x) to be heavy-tailed or light-tailed. Moreover, we introduce two subclasses of heavy-tailed distributions, i.e., ℳ and ℳ*, where ℳ contains almost all the most important heavy-tailed distributions in the literature. Some further discussions on the closure properties of ℳ and ℳ* under convolution are given, showing that both of them are ideal heavy-tailed subclasses. In the paper we also study the model of independent difference ξ = Z − θ, where Z and θ are two independent and non-negative random variables. We give intimate relationships of the tail distributions of ξ and Z, as well as relationships of tails of their corresponding equilibrium distributions. As applications, we apply the properties of class ℳ to risk theory. In the final, some miscellaneous problems and examples are laid, showing the complexity of characterizations on heavy-tailed distributions by means of r_e(x).

Keywords

equilibrium distribution / hazard function / equilibrium hazard rate (E.H.R.) / class ℳ and class ℳ* / heavy-tail / closure properties / convolution / independent difference / ladder height / ruin probability

References

[1] Embrechts P., Omey E. A property of long-tailed distribution. J Appl Probab, 1984, 21: 80-87.

[2] Embrechts P., Klüpplberg C., Mikosch T. Modelling Extremal Events for Insurance and Finance, 1997, Berlin: Springer.

[3] Chistyakov V. P. A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Probab Appl, 1964, 9: 640-648.

[4] Chover J., Ney P., Waiger S. Functions of probability measures. J Analyse Math, 1972, 26: 255-302.

[5] Borovkov A. A. Stochastic Processes in Queueing, 1976, New York: Springer.

[6] Pakes A. G. On the tails of waiting time distributions. J Appl Probab, 1975, 12: 555-564.

[7] Asmussen S. Applied Probability and Queues, 1987, Chichester: Wiley.

[8] Feller W. An Introduction to Probability Theory and Its Applications. Vol 2, 1971 2nd ed. New York: Wiley.

[9] Embrechts P., Veraverbeke N. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Math Econom, 1982, 1: 55-72.

[10] Veraverbeke N. Asymptotic behavior of Wiener-Hopf factors of a random walk. Stoch Proc Appl, 1977, 5: 27-37.

[11] Asmussen S. Ruin Probabilities, 2000, Singapore: World Scientific.

[12] Schmidli H. Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion. Insurance: Math Econom, 2001, 28: 13-20.

[13] Su C., Tang Q. H. Characterizations on heavy tailed distributions by means of hazard rate. Acta Mathematicae Applicatase Sinica, English Series, 2003, 19: 135-142.

[14] Su C., Tang Q. H. Lai T. L., Yang H., Yung S. P. Heavy-tailed distributions and their applications. Probability, Finance and Insurance: Proceedings of a workshop at the University of Hong Kong, 2004, Singapore: World Scientific, 218-236.

[15] Pitman E. J. G. Subexponential distribution functions. J Austral Math Soc, Ser A, 1980, 29: 337-347.

[16] Rolski T., Schmidli H., Schmidt V. Stochastic Processes for Insurance and Finance, 1999, Chichester: John Wiley and Sons.

[17] Klüpplberg C. Subexponential distributions and integrated tails. J Appl Probab, 1988, 25: 132-141.

[18] Björk T, Grandell J. An insensitivity property of the ruin probability. Scand Act J, 1985: 148–156

[19] Grandell J. Finite time ruin probabilities and martingales. Informatic, 1991, 2: 3-32.

[20] Kalashnikov V. Two-sided bounds of ruin probabilities. Scand Act J, 1996: 1–18

[21] Kalashnikov V. Geometric Sums: Bounds for Rate Event with Applications. Kluwer, 1997

[22] Frenz M., Schmidt V. An insensitivity property of ladder height distributions. J Appl Probab, 1992, 29: 695-712.

[23] Miyazawa M., Schmidt V. On ladder height distributions of general risk processes. Ann Appl Probab, 1993, 3: 763-776.

[24] Asmussen S., Schmidt V. The ascending ladder height distribution for a class of dependent random walks. Statistica Neerlnadica, 1993, 47: 1-9.

[25] Asmussen S., Schmidt V. Ladder height distributions with marks. Stoch Proc Appl, 1995, 58: 105-119.

[26] Wang Y. B., Wang K. Y. Asymptotics of the density of the supreum of a random walk with heavy-tailed increments. J Appl Probab, 2006, 43: 874-879.

[27] Korshunov D. On distribution tail of the maximum of a random walk. Stochastic Processes and Their Applications, 1997, 72: 97-103.

[28] Goldie C. M., Klüppelberg C. Adler R., Feldman R., Taqqu M. S. Subexponential distributions. A Practical Guide to Heavy Tails: Statistical Techniques and Applications, 1998, Boston: Birkhäuser.

[29] Su C., Chen Y. Behaviors of the product of independent random variables 1. Int Journal of Math Analysis, 2007, 1: 21-35.

[30] Haan L de. On Regular Variation and Its Application to Weak Convergence of Sample. Extremes, CWI Tract 32. Amsterdam, 1970

[31] Seneta E. Functions of Regular Variation, 1976, New York: Springer.