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Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure

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  • Leipus, Remigijus
  • Paukštys, Saulius
  • Šiaulys, Jonas

Abstract

In this paper we consider the sum Snξ≔ξ1+…+ξn of (possibly dependent and nonidentically distributed) real-valued random variables ξ1,…,ξn with dominatedly varying distributions. Assuming that the ξk’s follow the dependence structure, similar to the asymptotic independence, we obtain the asymptotic lower and upper bounds for the tail moment E((Snξ)m1{Snξ>x}), where m is a nonnegative integer, improving the bounds of Leipus et al. (2019). We also consider the case of nonnegative random variables. Using the obtained results, we get the asymptotic estimations for the Haezendonck–Goovaerts risk measure in two examples of sums with regularly varying and dominatedly varying (but not regularly varying) increments.

Suggested Citation

  • Leipus, Remigijus & Paukštys, Saulius & Šiaulys, Jonas, 2021. "Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 170(C).
    • Handle: RePEc:eee:stapro:v:170:y:2021:i:c:s0167715220303011
    DOI: 10.1016/j.spl.2020.108998
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    1. Shijie Wang & Yiyu Hu & LianQiang Yang & Wensheng Wang, 2018. "Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(20), pages 5054-5063, October.
    2. Xing-Fang Huang & Ting Zhang & Yang Yang & Tao Jiang, 2017. "Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks," Risks, MDPI, vol. 5(1), pages 1-14, March.
    3. Hua, Lei & Joe, Harry, 2011. "Second order regular variation and conditional tail expectation of multiple risks," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 537-546.
    4. Tang, Qihe & Yang, Fan, 2014. "Extreme value analysis of the Haezendonck–Goovaerts risk measure with a general Young function," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 311-320.
    5. Cline, D. B. H. & Samorodnitsky, G., 1994. "Subexponentiality of the product of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 75-98, January.
    6. Dickson,David C. M., 2016. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9781107154605, September.
    7. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
    8. Li, Jinzhu, 2013. "On pairwise quasi-asymptotically independent random variables and their applications," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2081-2087.
    9. Shijie Wang & Cen Chen & Xuejun Wang, 2017. "Some novel results on pairwise quasi-asymptotical independence with applications to risk theory," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(18), pages 9075-9085, September.
    10. Li Zhu & Haijun Li, 2012. "Asymptotic Analysis of Multivariate Tail Conditional Expectations," North American Actuarial Journal, Taylor & Francis Journals, vol. 16(3), pages 350-363.
    11. Tang, Qihe & Tsitsiashvili, Gurami, 2003. "Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 299-325, December.
    12. 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.
    13. 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.
    14. 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.
    15. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Zhao, Shouqi, 2015. "On finite-time ruin probabilities in a generalized dual risk model with dependence," European Journal of Operational Research, Elsevier, vol. 242(1), pages 134-148.
    16. Haezendonck, J. & Goovaerts, M., 1982. "A new premium calculation principle based on Orlicz norms," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 41-53, January.
    17. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.
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