Asymptotics of convolution with the semi-regular-variation tail and its application to risk
Tóm tắt
In this paper, according to a certain criterion, we divide the exponential distribution class into some subclasses. One of them is closely related to the regular-variation-tailed distribution class, and is called the semi-regular-variation-tailed distribution class. The new class possesses several nice properties, although distributions in it are not convolution equivalent. We give the precise tail asymptotic expression of convolutions of these distributions, and prove that the class is closed under convolution. In addition, we do not need to require the corresponding random variables to be identically distributed. Finally, we apply these results to a discrete time risk model with stochastic returns, and obtain the precise asymptotic estimation of the finite time ruin probability.
Tài liệu tham khảo
Asmussen, S., Foss, S., Korshunov, D.: Asymptotics for sums of random variables with local subexponential behavior. J. Theor. Probab 16, 489–518 (2003)
Bertoin, J., Doney, R.A.: Some asymptotic results for transient random walks. Adv. Appl. Probab. 28, 207–226 (1996)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Chen, Y.: The finite-time ruin probability with dependent insurance and financial risks. J. Appl. Probab. 48(4), 1035–1048 (2011)
Chistyakov, V.P.: A theorem on sums of independent positive random variables and its application to branching processes. Theory Probab. Appl 9, 640–648 (1964)
Chover, J., Ney, P., Wainger, S.: Functions of probability measures. J. Anal. Math 26, 255–302 (1973a)
Chover, J., Ney, P., Wainger, S.: Degeneracy properties of subcritical branching processes. Ann. Probab 1, 663–673 (1973b)
Cline, D.B.H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Their Appl 49, 75–98 (1994)
Embrechts, P., Goldie, C.M.: On closure and factorization properties of subexponential and related distributions. J. Aust. Math. Soc., Ser. A 2, 243–256 (1980)
Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Process. Appl. 13, 263–278 (1982)
Embrechts, P.C., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer (1997)
Foss, S., Korshunov, D.: Lower limits and equivalences for convolution tails. Ann. Probab. 1, 366–383 (2007)
Foss, S., Korshunov, D., Zachary, S.: An Introduction to Heavy-tailed and Subexponential Distributions. Springer, 2nd edn (2013)
Hashorva, E., Li, J.: ECOMOR and LCR reinsurance with gamma-like claims. Insur. Math. Econ. 53, 206–215 (2013)
Hashorva, E., Li, J.: Asymptotics for a discrete-time risk model with the emphasis on financial risk. Probab. Eng. Inform. Sci. 28(4), 573–588 (2014)
Klüppelberg, C.: Subexponential distributions and characterization of related classes. Probab. Theory Related Fields 82, 259–269 (1989)
Leslie, J.: On the non-closure under convolution of the subexponential family. J. Appl. Probab. 26, 58–66 (1989)
Li, J., Tang, Q.: Interplay of insurance and financial risks in a discrete-time model with strongly regular variation. Bernoulli 21(3), 1800–1823 (2015)
Lin, J., Wang, Y.: New examples of heavy-tailed O-subexponential distributions and related closure properties. Statist. Probab. Lett. 82, 427–432 (2012)
Murphree, E.S.: Some new results on the subexponential class. J. Appl. Probab. 26, 892–897 (1989)
Omey, E., Gulck, S.V., Vesilo, R.: Semi-heavy tails. Accepted by Lithuanian Mathematical Journal (2017)
Pakes, A.G.: Convolution equivalence and infinite divisibility. J. Appl. Probab. 41, 407–424 (2004)
Pitman, E.J.G.: Subexponential distribution functions. J. Austral. Math. Soc., Ser. A 29, 337–347 (1980)
Tang, Q.: Asymptotic ruin probabilities in finite horizon with subexponential losses and associated discount factors. Probab. Engrg. Inform. Sci. 20(1), 103–113 (2006)
Tang, Q.: From light tails to heavy tails through multiplier. Extremes 11, 379–391 (2008)
Tang, Q., Tsitsiashvili, G.: Precise estimates for the ruin probability in finite horizon in a discretetime model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108(2), 299–325 (2003)
Tang, Q., Tsitsiashvili, G.: Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Probab. 36(4), 1278–1299 (2004)
Wang, Y., Xu, H., Cheng, D., Yu, C.: The local asymptotic estimation for the supremum of a random walk. Statist. Papers 59, 99–126 (2018)
Watanabe, T.: Convolution equivalence and distributions of random sums. Probab. Theory Relat. Fields 142, 367–397 (2008)
Yang, Y., Wang, Y.: Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16(1), 55–74 (2013)
Xu, H., Cheng, F., Wang, Y., Cheng, D.: A necessary and sufficient condition for the subexponentiality of product convolution. Adv. Appl. Probab. 50 (1), 57–73 (2018)