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Approximation of the Tail Probability of Dependent Random Sums Under Consistent Variation and Applications

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  • Zhengyan Lin

    (Zhejiang University)

  • Xinmei Shen

    (Zhejiang University
    Dalian University of Technology)

Abstract

In this paper, we consider the random sums of one type of asymptotically quadrant sub-independent and identically distributed random variables {X, X i , i = 1, 2, ⋯ } with consistently varying tails. We obtain the asymptotic behavior of the tail $\textsf{P}(X_1+\cdots+X_\eta>x)$ under different cases of the interrelationships between the tails of X and η, where η is an integer-valued random variable independent of {X, X i , i = 1, 2, ⋯ }. We find out that the asymptotic behavior of $\textsf{P}(X_1+\cdots+X_\eta>x)$ is insensitive to the dependence assumed in the present paper. We state some applications of the asymptotic results to ruin probabilities in the compound renewal risk model under dependent risks. We also state some applications to a compound collective risk model under the Markov environment.

Suggested Citation

  • Zhengyan Lin & Xinmei Shen, 2013. "Approximation of the Tail Probability of Dependent Random Sums Under Consistent Variation and Applications," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 165-186, March.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:1:d:10.1007_s11009-011-9232-0
    DOI: 10.1007/s11009-011-9232-0
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    References listed on IDEAS

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    1. Ng, K.W. & Tang, Q.H. & Yang, H., 2002. "Maxima of Sums of Heavy-Tailed Random Variables," ASTIN Bulletin, Cambridge University Press, vol. 32(1), pages 43-55, May.
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    3. Xin-mei Shen & Zheng-yan Lin & Yi Zhang, 2009. "Uniform Estimate for Maximum of Randomly Weighted Sums with Applications to Ruin Theory," Methodology and Computing in Applied Probability, Springer, vol. 11(4), pages 669-685, December.
    4. Zhang, Yi & Shen, Xinmei & Weng, Chengguo, 2009. "Approximation of the tail probability of randomly weighted sums and applications," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 655-675, February.
    5. Cossette, Helene & Landriault, David & Marceau, Etienne, 2004. "Compound binomial risk model in a markovian environment," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 425-443, October.
    6. Tang, Qihe & Su, Chun & Jiang, Tao & Zhang, Jinsong, 2001. "Large deviations for heavy-tailed random sums in compound renewal model," Statistics & Probability Letters, Elsevier, vol. 52(1), pages 91-100, March.
    7. Cossette, Helene & Marceau, Etienne, 2000. "The discrete-time risk model with correlated classes of business," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 133-149, May.
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    Cited by:

    1. Shijie Wang & Yueli Yang & Yang Liu & Lianqiang Yang, 2023. "Asymptotics for a Bidimensional Renewal Risk Model with Subexponential Main Claims and Delayed Claims," Methodology and Computing in Applied Probability, Springer, vol. 25(3), pages 1-13, September.

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