Skip to main content

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

Abstract

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given.

This is a preview of subscription content, access via your institution.

References

  • H. Albrecher, S. Asmussen, and D. Kortschak, “Tail asymptotics for the sum of two heavy-tailed dependent risks,” Extremes vol. 9(2) pp. 107–130, 2006.

    MATH  Article  MathSciNet  Google Scholar 

  • S. Alink, M. Löwe, and M. V. Wüthrich, “Diversification of aggregate dependent risks,” Insurance. Mathematics & Economics vol. 35(1) pp. 77–95, 2004.

    MATH  Article  MathSciNet  Google Scholar 

  • S. Alink, M. Löwe, and M. V. Wüthrich, “Analysis of the expected shortfall of aggregate dependent risks,” Astin Bulletin vol. 35(1) pp. 25–43, 2005a

    MATH  Article  MathSciNet  Google Scholar 

  • S. Alink, M. Löwe, and M. V. Wüthrich, “Analysis of the diversification effect of aggregate dependent risks,” Statistica Neerlandica, 2005b (preprint).

  • S. Asmussen, and L. Rojas-Nandayapa, Sums of Dependent Lognormal Random Variables: Asymptotics and Simulation, Technical report. Thiele Center, 2006.

  • P. Barbe, A.-L. Fougéres, and C. Genest, “On the tail behavior of sums of dependent risks,” Astin Bulletin vol. 36(2) pp. 361–373, 2006.

    Article  MathSciNet  Google Scholar 

  • B. Basrak, The Sample Autocorrelation Function of Non-linear Time Series, Dissertation. University of Groningen, 2000.

  • B. Basrak, R. A. Davis, and T. Mikosch, “A characterization of multivariate regular variation,” Annals of Applied Probabability vol. 12(3) pp. 908–920, 2002.

    MATH  Article  MathSciNet  Google Scholar 

  • N. Bäuerle, and A. Müller, “Modeling and comparing dependencies in multivariate risk portfolios,” ASTIN Bulletin vol. 28(1) pp. 59–76, 1998.

    MATH  Article  Google Scholar 

  • J. Beirlant, Y. Goegebeur, J. Segers, and J. Teugels, Statistics of Extremes. Wiley: Chichester, 2004.

    MATH  Book  Google Scholar 

  • S. Demarta, and A. J. McNeil, “The t copula and related copulas,” Internatational Statististical Review vol. 73(1) pp. 111–129, 2005.

    MATH  Google Scholar 

  • P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events, Springer: Berlin, 1997.

    MATH  Google Scholar 

  • E. W. Frees, and E. A. Valdez, “Understanding relationships using copulas,” North American Actuarial Journal vol. 2(1) pp. 1–25, 1998.

    MATH  MathSciNet  Google Scholar 

  • J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Robert E. Krieger Publishing Co. Inc.: Melbourne, FL, 1987.

    MATH  Google Scholar 

  • J. L. Geluk, and L. de Haan, Regular Variation, Extensions and Tauberian Theorems, vol. 40 of CWI Tract. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987.

  • H. Hult, and F. Lindskog, “Multivariate extremes, aggregation and dependence in elliptical distributions,” Advances in Applied Probability vol. 34(3) pp. 587–608, 2002.

    MATH  Article  MathSciNet  Google Scholar 

  • H. Hult, and F. Lindskog, “On Kesten’s counterexample to the Cramér–Wold device for regular variation,” Bernoulli vol. 12(1) pp. 133–142, 2006.

    MATH  MathSciNet  Google Scholar 

  • H. Joe, Multivariate Models and Dependence Concepts, Chapman & Hall: London, 1997.

    MATH  Google Scholar 

  • O. Kallenberg, Random Measures. Akademie-Verlag: Berlin, 1983.

    MATH  Google Scholar 

  • S. A. Klugman, and R. Parsa, “Fitting bivariate loss distributions with copulas,” Insurance. Mathematics Economics vol. 24(1–2) pp. 139–148, 1999.

    MATH  Article  MathSciNet  Google Scholar 

  • C. Klüppelberg, and T. Mikosch, “Large deviations of heavy-tailed random sums with applications in insurance and finance,” J. Appl. Probab. vol. 34(2) pp. 293–308, 1997.

    MATH  Article  MathSciNet  Google Scholar 

  • Y. Malevergne, and D. Sornette, Extreme Financial Risks—from Dependence to Risk Management, Springer: Berlin, 2006.

    MATH  Google Scholar 

  • A. W. Marshall, and I. Olkin, “Domains of attraction of multivariate extreme value distributions,” Annals of Probability vol. 11(1) pp. 168–177, 1983.

    MATH  Article  MathSciNet  Google Scholar 

  • R. B. Nelsen, An Introduction to Copulas, Springer: New York, 1999.

    MATH  Google Scholar 

  • S. I. Resnick, Extreme Values, egular Variation, and Point Processes, Springer: New York, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hansjörg Albrecher.

Additional information

Dominik Kortschak was supported by the Austrian Science Fund Project P18392.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kortschak, D., Albrecher, H. Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables. Methodol Comput Appl Probab 11, 279–306 (2009). https://doi.org/10.1007/s11009-007-9053-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-007-9053-3

Keywords

  • Subexponential tail
  • Dependence
  • Copula
  • Multivariate regular variation
  • Maximum domain of attraction

AMS 2000 Subject Classification

  • 60G70
  • 62E20
  • 62H20
  • 62P05