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A necessary and sufficient condition for the subexponentiality of the product convolution

Part of: Distribution theory Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  20 March 2018

Hui Xu ,
Fengyang Cheng ,
Yuebao Wang  and
Dongya Cheng
Hui Xu*
Affiliation:
Soochow University
Fengyang Cheng*
Affiliation:
Soochow University
Yuebao Wang*
Affiliation:
Soochow University
Dongya Cheng*
Affiliation:
Soochow University
*
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
* Postal address: School of Mathematical Sciences, Soochow University, Suzhou, 215006, China.
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Abstract

Let X and Y be two independent and nonnegative random variables with corresponding distributions F and G. Denote by H the distribution of the product XY, called the product convolution of F and G. Cline and Samorodnitsky (1994) proposed sufficient conditions for H to be subexponential, given the subexponentiality of F. Relying on a related result of Tang (2008) on the long-tail of the product convolution, we obtain a necessary and sufficient condition for the subexponentiality of H, given that of F. We also study the reverse problem and obtain sufficient conditions for the subexponentiality of F, given that of H. Finally, we apply the obtained results to the asymptotic study of the ruin probability in a discrete-time insurance risk model with stochastic returns.

Keywords

Necessary and sufficient conditionproduct convolutionsubexponential distributiondiscrete-time insurance risk model

MSC classification

Primary: 60E05: Distributions
Secondary: 62E20: Asymptotic distribution theory 60G50: Sums of independent random variables; random walks
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 57 - 73
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1] Arendarczyk, M. and Dębicki, K. (2011). Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli 17, 194–210. Google Scholar
[2] Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ. Google Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. Google Scholar
[4] Chen, Y. (2011). The finite-time ruin probability with dependent insurance and financial risks. J. Appl. Prob. 48, 1035–1048. Google Scholar
[5] Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Prob. Appl. 9, 640–648. CrossRefGoogle Scholar
[6] Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49, 75–98. CrossRefGoogle Scholar
[7] Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256. Google Scholar
[8] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer, Berlin. Google Scholar
[9] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York. Google Scholar
[10] Foss, S., Korshunov, D. and Zachary, S. (2013). An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer, New York. Google Scholar
[11] Goovaerts, M. J. et al. (2005). The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuarial J. 2005, 446–461. Google Scholar
[12] Huang, W., Weng, C. and Zhang, Y. (2014). Multivariate risk models under heavy-tailed risks. Appl. Stoch. Models Business Industry 30, 341–360. Google Scholar
[13] Leslie, J. R. (1989). On the nonclosure under convolution of the subexponential family. J. Appl. Prob. 26, 58–66. Google Scholar
[14] Li, J. and Tang, Q. (2015). Interplay of insurance and financial risks in a discrete-time model with strongly regular variation. Bernoulli 21, 1800–1823. Google Scholar
[15] Lin, J. and Wang, Y. (2012). New examples of heavy-tailed O-subexponential distributions and related closure properties. Statist. Prob. Lett. 82, 427–432. CrossRefGoogle Scholar
[16] Liu, Y. and Tang, Q. (2010). The subexponential product convolution of two Weibull-type distributions. J. Austral. Math. Soc. 89, 277–288. Google Scholar
[17] Murphree, E. S. (1989). Some new results on the subexponential class. J. Appl. Prob. 26, 892–897. Google Scholar
[18] Nyrhinen, H. (1999). On the ruin probabilities in a general economic environment. Stoch. Process. Appl. 83, 319–330. CrossRefGoogle Scholar
[19] Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265–285. Google Scholar
[20] Pitman, E. J. G. (1980). Subexponential distribution functions. J. Austral. Math. Soc. Ser. A 29, 337–347. Google Scholar
[21] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. CrossRefGoogle Scholar
[22] Samorodnitsky, G. and Sun, J. (2016). Multivariate subexponential distributions and their applications. Extremes 19, 171–196. Google Scholar
[23] Tang, Q. (2006). The subexponentiality of products revisited. Extremes 9, 231–241. Google Scholar
[24] Tang, Q. (2008). From light tails to heavy tails through multiplier. Extremes 11, 379–391. Google Scholar
[25] Tang, Q. and Tsitsiashvili, G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Appl. 108, 299–325. Google Scholar
[26] Tang, Q. and Tsitsiashvili, G. (2004). Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Prob. 36, 1278–1299. Google Scholar
[27] Wang, Y., Xu, H., Cheng, D. and Yu, C. (2016). The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands. Statist. Papers 10.1007/s00362-016-0754-y. Google Scholar
[28] Xu, H., Foss, S. and Wang, Y. (2015). Convolution and convolution-root properties of long-tailed distributions. Extremes 18, 605–628. Google Scholar
[29] Xu, H., Scheutzow, M., Wang, Y. and Cui, Z. (2016). On the structure of a class of distributions obeying the principle of a single big jump. Prob. Math. Statist. 36, 121–135. Google Scholar
[30] Yang, Y. and Wang, Y. (2013). Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16, 55–74. CrossRefGoogle Scholar
[31] Zhang, Y., Shen, X. and Weng, C. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stoch. Process. Appl. 119, 655–675. Google Scholar