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Dindienė, Lina; Leipus, Remigijus; Šiaulys, Jonas

Closure property and tail probability asymptotics for randomly weighted sums of dependent random variables with heavy tails. (English) Zbl 1401.60028

J. Korean Math. Soc. 54, No. 6, 1879-1903 (2017).
Summary: In this paper we study the closure property and probability tail asymptotics for randomly weighted sums \(S_n^\Theta=\Theta_1 X_1 + \cdots+\Theta_n X_n\) for long-tailed random variables \(X_1,\dots,X_n\) and positive bounded random weights \(\Theta_1,\dots,\Theta_n\) under similar dependence structure as in [Y. Yang et al., Stat. Probab. Lett. 91, 162–170 (2014; Zbl 1288.62078)]. In particular, we study the case where the distribution of random vector \((X_1,\dots,X_n)\) is generated by an absolutely continuous copula.

Cited in 2 Documents

MSC:

60E99 Distribution theory
62E20 Asymptotic distribution theory in statistics
62H20 Measures of association (correlation, canonical correlation, etc.)

Keywords:

randomly weighted sum; long-tail distribution; copula; FGM copula

Citations:

Zbl 1288.62078
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Full Text: DOI

References:

[1] 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. · Zbl 1224.91041
[2] 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. · Zbl 1054.60012
[3] 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. · Zbl 1232.62136
[4] 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. · Zbl 1181.62011
[5] 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. 1902L. DINDIEN ˙E, R. LEIPUS, AND J. ˇSIAULYS · Zbl 0425.60011
[6] S. Foss, D. Korshunov, and S. Zachary, Convolutions of long-tailed and subexponential distributions, J. Appl. Probab. 46 (2009), no. 3, 756-767. · Zbl 1178.60011
[7] 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. · Zbl 1294.60067
[8] J. Geluk and K. W. Ng, Tail behavior of negatively associated heavy-tailed sums, J. Appl. Probab. 43 (2006), no. 2, 587-593. · Zbl 1104.60313
[9] J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theoret. Probab. 22 (2009), no. 4, 871-882. · Zbl 1177.62017
[10] 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. · Zbl 1348.91155
[11] 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. · Zbl 1319.62206
[12] J. R. Leslie, On the non-closure under convolution of the subexponential family, J. Appl. Probab. 26 (1989), no. 1, 58-66. · Zbl 0672.60027
[13] J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Statist. Probab. Lett. 83 (2013), no. 9, 2081-2087. · Zbl 1279.62212
[14] 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. · Zbl 1205.62061
[15] 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. · Zbl 1209.62246
[16] 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. · Zbl 1242.91094
[17] K. W. Ng, Q. Tang, and H. Yang, Maxima of sums of heavy-tailed random variables, Astin Bull. 32 (2002), no. 1, 43-55. · Zbl 1098.60505
[18] 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. · Zbl 1075.91563
[19] , Randomly weighted sums of subexponential random variables with application to ruin theory, Extremes 6 (2003), no. 3, 171-188. · Zbl 1049.62017
[20] Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes 17 (2014), no. 3, 467-493. · Zbl 1328.62089
[21] K. Wang, Randomly weighted sums of dependent subexponential random variables, Lith. Math. J. 51 (2011), no. 4, 573-586. · Zbl 1322.60076
[22] 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. · Zbl 1263.91027
[23] T. Watanabe and K. Yamamuro, Ratio of the tail of an infinitely divisible distribution on the line to that of its L´evy measure, Electron. J. Probab. 15 (2010), no. 2, 44-74. · Zbl 1193.60023
[24] H. Xu, S. Foss, and Y. Wang, Convolution and convolution-root properties of long-tailed distributions, Extremes 18 (2015), no. 4, 605-628. · Zbl 1327.60042
[25] Y. Yang, R. Leipus, and J. ˇSiaulys, Tail probability of randomly weighted sums of subexponential random variables under a dependence structure, Statist. Probab. Lett. 82 (2012), no. 9, 1727-1736. · Zbl 1334.62029
[26] , Closure property and maximum of randomly weighted sums with heavy tailed increments, Statist. Probab. Lett. 91 (2014), 162-170. · Zbl 1288.62078
[27] C. Zhang, Uniform asymptotics for the tail probability of weighted sums with heavy tails, Statist. Probab. Lett. 94 (2014), 221-229. · Zbl 1381.62045
[28] 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. · Zbl 1171.62011
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