On two broad classes of heavy-tailed distributions


Chun Su, Zhishui Hu

2004, 2, . 195204


Since the introduction of Class $\mathcal{M}$ and Class $\mathcal{M^?}$, they have played important roles in insurance to describe tail equivalence of ruin probability and tail behavior of the deficit at ruin. And in insurance and finance most of heavy-tailed distributions with finite and positive expectations belong to Class $\mathcal{M}$. So it is important to study tail behaviors of Class $\mathcal{M}$ and Class $\mathcal{M^?}$. In this paper, we obtain some results on essential tail behaviors of these two classes.

:
class $\mathcal{M}$, class $\mathcal{M^?}$, class $\mathcal{D}$, tail behaviors

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[1] S. Asmussen, Ruin Probabilities, World Scientic, Singapore, 2000.
[2] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
[3] P. Embrechts, C. Kluppelberg, T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997.
[4] M. Greiner, M. Jobmman, C. Kluppelberg, (1999): Telecommunication Traffic, Queueing Models and Subexponential Distributions, Queueing Systems, 33 (1999), 125152.
[5] P. R. Jelenkovic, A. A. Lazar, Asymptotic results for multiplexing subexponential on-off processes, Adv. Appl. Prob., 31 (1999), 394421.
[6] V. Kalashinikov, Geometric Sums Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
[7] T. Mikosch, A. Nagaev, Rates in Approximations to Ruin Probabiliyies for Heavy Tailed Distributions, Extremes, 4:1 (2001), 6778.
[8] K. Ng, Q. Tang, H. Yang, Maxima of sums of heavy-tailed random variables, Astin Bull., 32:1 (2002), 4355.
[9] K. Ng, Q. Tang, J. Yan, H. Yang, Precise large deviations for the prospective-loss process, J. Appl. Probab., 40:2 (2003), 391400.
[10] K. Ng, Q. Tang, J. Yan, H. Yang, Precise large deviations for sums of random variables with consistently varying tails, Adv. in Appl. Probab., 41:1 (2004), 93107.
[11] T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic processes for insurance and finance, John Wiley & Sons, Chichester, England, 1999.
[12] S. Schlegel, Ruin probabilities in perturbed risk models, The interplay between insurance, finance and control (Aarhus, 1997), Insurance Math. Econom., 22, no. 1, 1998, 93104.
[13] C. Su, Z. S. Hu, Q. H. Tang, Characterizations on heaviness of distribution tails of non-negative variables, Adv. Math. (China), 32 (2003), 606614.
[14] Chun Su, Qihe Tang, Characterizations on Heavy-tailed Distributions by Means of Hazard Rate, Acta Mathematicae Applicatae Sinica, English Series, 19:1 (2003), 135142.
[15] Chun Su, Qihe Tang, Yu Chen, Hanying Liang, Two Broad Classes of Heavy-Tailed Distributions and Their Applications to Insurance (to appear).
[16] Q. H. Tang, Moments of the deficit at ruin in the renewal model with heavy-tailed claims, Science in China (Series A), 2002 (to appear).