Advertisement

Lithuanian Mathematical Journal

, Volume 57, Issue 1, pp 142–153 | Cite as

Asymptotics for the partial sum and its maximum of dependent random variables*

  • Ting Zhang
  • Xi-Nian Fang
  • Jie Liu
  • Yang Yang
Article

Abstract

Let X 1,…,X n be pairwise asymptotically independent or pairwise upper extended negatively dependent real-valued random variables. Under the condition that the distribution of the maximum of X 1,…,X n belongs to some subclass of heavy-tailed distributions, we investigate the asymptotic behavior of the partial sum and its maximum generated by dependent X 1,…,X n . As an application, we consider a discrete-time risk model with insurance and financial risks and derive the asymptotics for the finite-time ruin probability.

Keywords

pairwise asymptotical independence pairwise upper extended negative dependence dominated variation long tail discrete-time risk model finite-time ruin probability 

MSC

60E05 60G70 62E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.CrossRefMATHGoogle Scholar
  2. 2.
    H.W. Block, T.H. Savits, and M. Shaked, Some concepts of negative dependence, Ann. Probab., 10:765–772, 1982.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Y. Chen and K.C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25:76–89, 2009.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    V.P. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching process, Theory Probab. Appl., 9:640–648, 1964.CrossRefGoogle Scholar
  5. 5.
    D.B.H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Processes Appl., 49:75–98, 1994.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    L. de Haan and A. Ferreira, Extreme Value Theory: An Introduction, Springer, New York, 2006.CrossRefMATHGoogle Scholar
  7. 7.
    L. Dindien˙e and R. Leipus, Weak max-sum equivalence for dependent heavy-tailed random variables, Lith. Math. J., 56:49–59, 2016.Google Scholar
  8. 8.
    N. Ebrahimi andM. Ghosh, Multivariate negative dependence, Commun. Stat., Theory Methods, 10:307–337, 1981.Google Scholar
  9. 9.
    P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin, 1997.CrossRefMATHGoogle Scholar
  10. 10.
    J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theor. Probab., 22:871–882, 2009.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    K. Joag-Dev and F. Proschan, Negative association of random variables with application, Ann. Stat., 11:286–295, 1983.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    H. Joe, Parametric families of multivariate distributions with given margins, J. Multivariate Anal., 46:262–282, 1993.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    B. Ko and Q. Tang, Sums of dependent nonnegative random variables with subexponential tails, J. Appl. Probab., 45:85–95, 2008.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    J. Li and Q. Tang, A note on max-sum equivalence, Stat. Probab. Lett., 80:1720–1723, 2010.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    L. Liu, Precise large deviations for dependent random variables with heavy tails, Stat. Probab. Lett., 79:1290–1298, 2009.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    X. Liu, Q. Gao, and Y. Wang, A note on a dependent risk model with constant interest rate, Stat. Probab. Lett., 82:707–712, 2012.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    R. Norberg, Ruin problems with assets and liabilities of diffusion type, Stochastic Processes Appl., 81:255–269, 1999.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    L. Peng and Q. Yao, Nonparametric regression under dependent errors with infinite variance, Ann. Inst. Stat. Math., 56:73–86, 2004.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    S.I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987.CrossRefMATHGoogle Scholar
  20. 20.
    Y. Yang, R. Leipus, and L. Dindien˙e, On the max-sum equivalence in presence of negative dependence and heavy tails, Information Technology and Control, 2:215–220, 2015.Google Scholar
  21. 21.
    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.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Y. Zhang, X. Shen, and C.Weng, Approximation of the tail probability of randomly weighted sums and applications, Stochastic Processes Appl., 119:655–675, 2009.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    B. Zwart, S. Borst, and M. Mandjes, Exact asymptotics for fluid queues fed by multiple heavy-tailed on–off flows, Ann. Appl. Probab., 14:903–957, 2004.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of ScienceNanjing Audit UniversityNanjingPR China
  2. 2.The School of ManagementUniversity of Science and Technology of ChinaHefeiPR China

Personalised recommendations