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Expectation of the truncated randomly weighted sums with dominatedly varying summands

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Abstract

We consider the asymptotic behavior of the values P(S > x), E(S 1{S>x}), and E(S | S > x). Here S = θ1X1 + θ2X2 + · · · + θnXn is a randomly weighted sum of the basic random variables X1,X2, . . . , Xn, which follow some special dependence structure, and {θ1, θ2, . . . , θn} is a collection of nonnegative and arbitrarily dependent random weights; the collections {X1,X2, . . .,Xn} and {θ1, θ2, . . . , θn} are supposed to be independent. We derive asymptotic formulas in the case where the number of summands n is fixed and the distributions of the basic random variables are dominatedly varying.We apply them to some values related to the risk measures of certain weighted sums.

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References

  1. C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking and Finance, 26:1487–1503, 2002.

    Article  Google Scholar 

  2. I.M. Andrulytė, M. Manstavičius, and J. Šiaulys, Randomly stopped maximum and maximum of sums with consistently varying distributions, Mod. Stoch., Theory Appl., 4:65–78, 2017.

    Article  MathSciNet  Google Scholar 

  3. A.V. Asimit, E. Furman, Q. Tang, and R. Vernic, Asymptotics for risk capital allocations based on conditional tail expectation, Insur. Math. Econ., 49:310–324, 2011.

    Article  MathSciNet  Google Scholar 

  4. H. Assa, M. Morales, and H. Omidi Firousi, On the capital allocation problem for a new coherent risk measure in collective risk theory, Risks, 4:30, 2016.

    Article  Google Scholar 

  5. M. Brandtner and W. Kürsten, Decision making with expected shortfall and spectral risk measures: The problem of comparative risk aversion, Journal of Banking and Finance, 58:268–280, 2015.

    Article  Google Scholar 

  6. Y. Chen, J. Liu, and F. Liu, Ruin with insurance and financial risks following the least risky FGM dependence structure, Insur. Math. Econ., 62:98–106, 2015.

    Article  MathSciNet  Google Scholar 

  7. Y. Chen and K.C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25:76–89, 2009.

    Article  MathSciNet  Google Scholar 

  8. D. Cheng, Randomly weighted sums of dependent random variables with dominated variation, J. Math. Anal. Appl., 420:1617–1633, 2014.

    Article  MathSciNet  Google Scholar 

  9. V.P. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching random processes, Theory Probab. Appl., 9:640–648, 1964.

    Article  MathSciNet  Google Scholar 

  10. S. Danilenko, J. Markevičiūtė, and J. Šiaulys, Randomly stopped sums with exponential-type distributions, Nonlinear Anal. Model. Control, 22:793–807, 2017.

    Article  MathSciNet  Google Scholar 

  11. J. Dhaene, A. Tsanakas, E.A. Valdez, and S. Vanduffel, Optimal capital allocation principles, The Journal of Risk and Insurance, 79:1–28, 2012.

    Article  Google Scholar 

  12. L. Dindienė and R. Leipus, A note on the tail behavior of randomlyweighted and stopped dependent sums, Nonlinear Anal. Model. Control, 20:263–273, 2015.

    Article  MathSciNet  Google Scholar 

  13. L. Dindienė and R. Leipus, Weak max-sum equivalence for dependent heavy-tailed random variables, Lith. Math. J., 56:49–59, 2016.

    Article  MathSciNet  Google Scholar 

  14. P. Embrechts, C. Klüppelberg, and T.Mikosch, Modelling Extremal Events for Insurance and Finance, Stoch.Model. Appl. Probab., Vol. 33, Springer, Berlin, Heidelberg, 1997.

  15. A.L. Fougeres and C. Mercadier, Risk measures and multivariate extensions of Breiman’s theorem, J. Appl. Probab., 49:364–384, 2012.

    Article  MathSciNet  Google Scholar 

  16. Q. Gao and Y. Wang, Randomly weighted sums with dominated varying-tailed increments and application to risk theory, J. Korean Stat. Soc., 39:305–314, 2010.

    Article  MathSciNet  Google Scholar 

  17. J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theor. Probab., 22:871–882, 2009.

    Article  MathSciNet  Google Scholar 

  18. C.M. Goldie, Subexponential distributions and dominated-variation tails, J. Appl. Probab., 15:440–442, 1978.

    Article  MathSciNet  Google Scholar 

  19. E. Hashorva and J. Li, Asymptotics for a discrete-time risk model with the emphasis on financial risk, Probab. Eng. Inf. Sci., 28:573–588, 2014.

    Article  MathSciNet  Google Scholar 

  20. L. Hua and H. Joe, Second order regular variation and conditional tail expectation of multiple risks, Insur. Math. Econ., 49:537–546, 2011.

    Article  MathSciNet  Google Scholar 

  21. X.F. Huang, T. Zhang, Y. Yang, and T. Jiang, Ruin probabilities in a dependent discrete-time risk model with Gamma-like tailed insurance risks, Risks, 5:14, 2017.

    Article  Google Scholar 

  22. P. Jordanova and M. Stehlik, Mixed Poisson process with Pareto mixing variable and its risk applications, Lith. Math. J., 56:189–206, 2016.

    Article  MathSciNet  Google Scholar 

  23. D.G. Konstantinides and C.E. Kountzakis, Risk measures in ordered normed linear spaces with non-empty cone interior, Insur. Math. Econ., 48:111–122, 2011.

    Article  MathSciNet  Google Scholar 

  24. J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Stat. Probab. Lett., 83:2081–2087, 2013.

    Article  MathSciNet  Google Scholar 

  25. R. Liu and D. Wang, The ruin probabilities of a discrete-time risk model with dependent insurance and financial risks, J. Math. Anal. Appl., 444:80–94, 2016.

    Article  MathSciNet  Google Scholar 

  26. X. Liu, Q. Gao, and Y. Wang, A note on a dependent risk model with constant interest rate, Stat. Probab. Lett., 82:707–712, 2011.

    Article  MathSciNet  Google Scholar 

  27. M. Mailhot and M. Mesfioui, Multivariate TVaR-based risk decomposition for vector-valued portfolios, Risks, 4:33, 2016.

    Article  Google Scholar 

  28. A.J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton Series in Finance, Princeton Univ. Press, Princeton, NJ, 2005.

    MATH  Google Scholar 

  29. H. Nyrhinen, On the ruin probabilities in a general economic environment, Stochastic Processes Appl., 83:319–330, 1999.

    Article  MathSciNet  Google Scholar 

  30. H. Nyrhinen, Finite and infinite time ruin probabilities in a stochastic economic environment, Stochastic Processes Appl., 92:265–285, 2001.

    Article  MathSciNet  Google Scholar 

  31. S.I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 1987.

    Book  Google Scholar 

  32. 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 Processes Appl., 108:299–325, 2003.

    Article  MathSciNet  Google Scholar 

  33. Q. Tang and Z. Yuan, A hybrid estimate for the finite-time ruin probability in a bivariate autoregressive risk model with application to portfolio optimization, N. Am. Actuar. J., 16:378–397, 2012.

    Article  MathSciNet  Google Scholar 

  34. Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17:467–493, 2014.

    Article  MathSciNet  Google Scholar 

  35. S. Wang, C. Chen, and X. Wang, Some novel results on pairwise quasi-asymptotical independence with applications to risk theory, Commun. Stat., Theory Methods, 46:9075–9085, 2017.

    Article  MathSciNet  Google Scholar 

  36. H. Yang, W. Gao, and J. Li, Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks, Scand. Actuar. J., 2016:1–17, 2016.

    MathSciNet  MATH  Google Scholar 

  37. Y. Yang, E. Ignatavičiūtė, and J. Šiaulys, Conditional tail expectation of randomly weighted sums with heavy-tailed distributions, Stat. Probab. Lett., 105:20–28, 2015.

    Article  MathSciNet  Google Scholar 

  38. Y. Yang and D.G. Konstantinides, Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks, Scand. Actuar. J., 2015:641–659, 2015.

    MathSciNet  MATH  Google Scholar 

  39. Y. Yang, R. Leipus, and J. Šiaulys, On the ruin probability in a dependent discrete time risk model with insurance and financial risks, J. Comput. Appl. Math., 236:3286–3295, 2012.

    Article  MathSciNet  Google Scholar 

  40. Y. Yang, R. Leipus, and J. Šiaulys, Asymptotics for randomly weighted and stopped dependent sums, Stochastics, 88:300–319, 2016.

    Article  MathSciNet  Google Scholar 

  41. Y. Yang and Y. Wang, Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims, Stat. Probab. Lett., 80:143–154, 2009.

    Article  MathSciNet  Google Scholar 

  42. Y. Yang and K.C. Yuen, Asymptotics for a discrete-time risk model with Gamma-like insurance risks, Scand. Actuar. J., 2016:565–579, 2016.

    Article  MathSciNet  Google Scholar 

  43. Y. Yang, T. Zhang, and K.C. Yuen, Approximations for finite-time ruin probability in a dependent discrete-time risk model with CMC simulations, J. Comput. Appl. Math., 321:143–159, 2017.

    Article  MathSciNet  Google Scholar 

  44. L. Yi, Y. Chen, and C. Su, Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation, J. Math. Anal. Appl., 376:365–372, 2011.

    Article  MathSciNet  Google Scholar 

  45. L. Zhu and H. Li, Asymptotic analysis of multivariate tail conditional expectations, N. Am. Actuar. J., 16:350–363, 2012.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Vilnius University, Naugarduko str. 24, LT-03225, Vilnius, Lithuania

    Eglė Jaunė & Jonas Šiaulys

  2. Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska str. 64, Kyiv, 01601, Ukraine

    Olena Ragulina

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  1. Eglė Jaunė
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  2. Olena Ragulina
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  3. Jonas Šiaulys
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Correspondence to Eglė Jaunė.

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The author is supported by grant No. S-MIP-17-72 from the Research Council of Lithuania.

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Jaunė, E., Ragulina, O. & Šiaulys, J. Expectation of the truncated randomly weighted sums with dominatedly varying summands. Lith Math J 58, 421–440 (2018). https://doi.org/10.1007/s10986-018-9408-1

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  • DOI: https://doi.org/10.1007/s10986-018-9408-1

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