Browse > Article
http://dx.doi.org/10.4134/JKMS.j160753

CLOSURE PROPERTY AND TAIL PROBABILITY ASYMPTOTICS FOR RANDOMLY WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES WITH HEAVY TAILS  

Dindiene, Lina (Faculty of Mathematics and Informatics Vilnius University)
Leipus, Remigijus (Faculty of Mathematics and Informatics Vilnius University)
Siaulys, Jonas (Faculty of Mathematics and Informatics Vilnius University)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.6, 2017 , pp. 1879-1903 More about this Journal
Abstract
In this paper we study the closure property and probability tail asymptotics for randomly weighted sums $S^{\Theta}_n={\Theta}_1X_1+{\cdots}+{\Theta}_nX_n$ for long-tailed random variables $X_1,{\ldots},X_n$ and positive bounded random weights ${\Theta}_1,{\ldots},{\Theta}_n$ under similar dependence structure as in [26]. In particular, we study the case where the distribution of random vector ($X_1,{\ldots},X_n$) is generated by an absolutely continuous copula.
Keywords
randomly weighted sum; long-tail distribution; copula; FGM copula;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. Li, Q. Tang, and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv. in Appl. Probab. 42 (2010), no. 4, 1126-1146.   DOI
2 J. Li and R. Wu, Asymptotic ruin probabilities of the renewal model with constant interest force and dependent heavy-tailed claims, Acta Math. Appl. Sin. Engl. Ser. 27 (2011), no. 2, 329-338.   DOI
3 X. Liu, Q. Gao, and Y. Wang, A note on a dependent risk model with constant interest rate, Statist. Probab. Lett. 82 (2012), no. 4, 707-712.   DOI
4 K. W. Ng, Q. Tang, and H. Yang, Maxima of sums of heavy-tailed random variables, Astin Bull. 32 (2002), no. 1, 43-55.   DOI
5 Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl. 108 (2003), no. 2, 299-325.   DOI
6 Q. Tang and G. Tsitsiashvili, Randomly weighted sums of subexponential random variables with application to ruin theory, Extremes 6 (2003), no. 3, 171-188.   DOI
7 Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes 17 (2014), no. 3, 467-493.   DOI
8 K. Wang, Randomly weighted sums of dependent subexponential random variables, Lith. Math. J. 51 (2011), no. 4, 573-586.   DOI
9 K. Wang, Y. Wang, and Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab. 15 (2013), no. 1, 109-124.   DOI
10 T. Watanabe and K. Yamamuro, Ratio of the tail of an infinitely divisible distribution on the line to that of its Levy measure, Electron. J. Probab. 15 (2010), no. 2, 44-74.
11 H. Xu, S. Foss, and Y. Wang, Convolution and convolution-root properties of long-tailed distributions, Extremes 18 (2015), no. 4, 605-628.   DOI
12 S. Foss, D. Korshunov, and S. Zachary, Convolutions of long-tailed and subexponential distributions, J. Appl. Probab. 46 (2009), no. 3, 756-767.   DOI
13 Y. Yang, R. Leipus, and J. Siaulys, Tail probability of randomly weighted sums of subexponential random variables under a dependence structure, Statist. Probab. Lett. 82 (2012), no. 9, 1727-1736.   DOI
14 Y. Yang, R. Leipus, and J. Siaulys, Closure property and maximum of randomly weighted sums with heavy tailed increments, Statist. Probab. Lett. 91 (2014), 162-170.   DOI
15 C. Zhang, Uniform asymptotics for the tail probability of weighted sums with heavy tails, Statist. Probab. Lett. 94 (2014), 221-229.   DOI
16 C. Zhu and Q. Gao, The uniform approximation of the tail probability of the randomly weighted sums of subexponential random variables, Statist. Probab. Lett. 78 (2008), no. 15, 2552-2558.   DOI
17 A. V. Asimit and A. L. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model, Scand. Actuar. J. 2010 (2010), no. 2, 93-104.   DOI
18 J. Cai and Q. Tang, On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications, J. Appl. Probab. 41 (2004), no. 1, 117-130.   DOI
19 Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models 25 (2009), no. 1, 76-89.   DOI
20 P. Embrechts and C. M. Goldie, On closure and factorization properties of subexponential and related distributions, J. Austral. Math. Soc. Ser. A 29 (1980), no. 2, 243-256.   DOI
21 Q. Gao and Y. Wang, Randomly weighted sums with dominated varying-tailed increments and application to risk theory, J. Korean Statist. Soc. 39 (2010), no. 3, 305-314.   DOI
22 J. Geluk and K. W. Ng, Tail behavior of negatively associated heavy-tailed sums, J. Appl. Probab. 43 (2006), no. 2, 587-593.   DOI
23 J. R. Leslie, On the non-closure under convolution of the subexponential family, J. Appl. Probab. 26 (1989), no. 1, 58-66.   DOI
24 J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theoret. Probab. 22 (2009), no. 4, 871-882.   DOI
25 T. Jiang, Y. Wang, Y. Chen, and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insurance Math. Econom. 64 (2015), 45-53.   DOI
26 F. Kong and G. Zong, The finite-time ruin probability for ND claims with constant interest force, Statist. Probab. Lett. 78 (2008), no. 17, 3103-3109.   DOI
27 J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Statist. Probab. Lett. 83 (2013), no. 9, 2081-2087.   DOI
28 Y. Chen, K. W. Ng, and K. C. Yuen, The maximum of randomly weighted sums with long tails in insurance and finance, Stoch. Anal. Appl. 29 (2011), no. 6, 1033-1044.   DOI