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Asymptotics of Random Contractions

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  • Enkelejd Hashorva
  • Anthony G. Pakes
  • Qihe Tang

Abstract

In this paper we discuss the asymptotic behaviour of random contractions $X=RS$, where $R$, with distribution function $F$, is a positive random variable independent of $S\in (0,1)$. Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of $X$ assuming that $F$ is in the max-domain of attraction of an extreme value distribution and the distribution function of $S$ satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.

Suggested Citation

  • Enkelejd Hashorva & Anthony G. Pakes & Qihe Tang, 2010. "Asymptotics of Random Contractions," Papers 1008.0126, arXiv.org.
    • Handle: RePEc:arx:papers:1008.0126
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    Cited by:

    1. Hashorva, Enkelejd & Li, Jinzhu, 2013. "ECOMOR and LCR reinsurance with gamma-like claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 206-215.
    2. Hashorva, Enkelejd, 2015. "Extremes of aggregated Dirichlet risks," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 334-345.
    3. Claude Lefèvre & Stéphane Loisel & Pierre Montesinos, 2020. "Bounding Basis-Risk Using s-convex Orders on Beta-unimodal Distributions," Post-Print hal-02611227, HAL.
    4. Qu, Zhihui & Chen, Yu, 2013. "Approximations of the tail probability of the product of dependent extremal random variables and applications," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 169-178.
    5. Hashorva, Enkelejd & Ling, Chengxiu & Peng, Zuoxiang, 2014. "Second-order tail asymptotics of deflated risks," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 88-101.
    6. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    7. N. Balakrishnan & A. Stepanov, 2014. "On the Use of Bivariate Mellin Transform in Bivariate Random Scaling and Some Applications," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 235-244, March.
    8. Tang, Qihe & Yang, Fan, 2012. "On the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 217-227.
    9. Popivoda, Goran & Stamatović, Siniša, 2019. "On probability of high extremes of Gaussian fields with a smooth random trend," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 29-35.
    10. Asimit, Alexandru V. & Furman, Edward & Tang, Qihe & Vernic, Raluca, 2011. "Asymptotics for risk capital allocations based on Conditional Tail Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 310-324.
    11. Chen, Yiqing & Liu, Jiajun & Liu, Fei, 2015. "Ruin with insurance and financial risks following the least risky FGM dependence structure," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 98-106.
    12. Ling, Chengxiu & Peng, Zuoxiang, 2016. "Tail asymptotics of generalized deflated risks with insurance applications," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 220-231.
    13. Mercè Claramunt, M. & Lefèvre, Claude & Loisel, Stéphane & Montesinos, Pierre, 2022. "Basis risk management and randomly scaled uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 123-139.
    14. Yang, Yang & Hashorva, Enkelejd, 2013. "Extremes and products of multivariate AC-product risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 312-319.
    15. Yang, Yingying & Hu, Shuhe & Wu, Tao, 2011. "The tail probability of the product of dependent random variables from max-domains of attraction," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1876-1882.
    16. Constantinescu, Corina & Hashorva, Enkelejd & Ji, Lanpeng, 2011. "Archimedean copulas in finite and infinite dimensions—with application to ruin problems," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 487-495.
    17. Jing Liu & Huan Zhang, 2017. "Asymptotic Estimates for the One-Year Ruin Probability under Risky Investments," Risks, MDPI, vol. 5(2), pages 1-11, May.

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