Siberian Mathematical Journal

, Volume 53, Issue 6, pp 965–983 | Cite as

Tail asymptotics for dependent subexponential differences

  • H. Albrecher
  • S. Asmussen
  • D. KortschakEmail author


We study the asymptotic behavior of ℙ(XY > u) as u → ∞, where X is subexponential, Y is positive, and the random variables X and Y may be dependent. We give criteria under which the subtraction of Y does not change the tail behavior of X. It is also studied under which conditions the comonotonic copula represents the worst-case scenario for the asymptotic behavior in the sense of minimizing the tail of XY. Some explicit construction of the worst-case copula is provided in other cases.


subexponential random variables differences dependence copulas mean excess function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albrecher H., Asmussen S., and Kortschak D., “Tail asymptotics for the sum of two heavy-tailed dependent risks,” Extremes, 9, No. 2, 107–130 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Mitra A. and Resnick S., “Aggregation of rapidly varying risks and asymptotic independence,” Adv. Appl. Probab., 41, No. 3, 797–828 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ko B. and Tang Q., “Sums of dependent nonnegative random variables with subexponential tails,” J. Appl. Probab., 45, No. 1, 85–94 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Kortschak D. and Albrecher H., “Asymptotic results for the sum of dependent non-identically distributed random variables,” Methodol. Comput. Appl. Probab., 11, No. 3, 279–306 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Foss S. and Richards A., “On sums of conditionally independent subexponential random variables,” Math. Oper. Res., 35, No. 1, 102–119 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Asmussen S. and Albrecher H., Ruin Probabilities (2nd ed.), World Scientific, Singapore (2010).zbMATHCrossRefGoogle Scholar
  7. 7.
    Albrecher H. and Teugels J., “Exponential behavior in the presence of dependence in risk theory,” J. Appl. Probab., 43, No. 1, 257–273 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Boudreault M., Cossette H., Landriault D., and Marceau E., “On a risk model with dependence between interclaim arrivals and claim sizes,” Scand. Actuar. J., No. 5, 265–285 (2006).Google Scholar
  9. 9.
    Asimit A. V. and Badescu A. L., “Extremes on the discounted aggregate claims in a time dependent risk model,” Scand. Actuar. J., No. 2, 93–104 (2010).Google Scholar
  10. 10.
    Li J., Tang Q., and Wu R., “Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model,” Adv. Appl Probab., 42, No. 4, 1126–1146 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Albrecher H. and Boxma O. J., “A ruin model with dependence between claim sizes and claim intervals,” Insurance Math. Econom., 35, No. 2, 245–254 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Asmussen S. and Biard R., “Ruin probabilities for a regenerative Poisson gap generated risk process,” Eur. Actuar. J., 1, No. 1, 3–22 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jelenković P., Momcilović P., and Zwart B., “Reduced load equivalence under subexponentiality,” Queueing Syst., 46, No. 1–2, 97–112 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cline D. B. H., “Intermediate regular and Π variation,” Proc. London Math. Soc., 68, No. 3, 594–616 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Robert C. Y. and Segers J., “Tails of random sums of a heavy-tailed number of light-tailed terms,” Insurance Math. Econom., 43, No. 1, 85–92 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Foss S., Korshunov D., and Zachary S., An Introduction to Heavy-Tailed and Subexponential Distributions, Springer-Verlag, New York (2011).zbMATHCrossRefGoogle Scholar
  17. 17.
    Resnick S. I., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York (1987).zbMATHGoogle Scholar
  18. 18.
    Galambos J., The Asymptotic Theory of Extreme Order Statistics, Robert E. Krieger Publ. Co. Inc., Melbourne, FL (1987).zbMATHGoogle Scholar
  19. 19.
    Bingham N. H., Goldie C. M., and Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge (1989) (Encyclopedia of Mathematics and its Applications; V. 27).zbMATHGoogle Scholar
  20. 20.
    Nelsen R. B., An Introduction to Copulas, Springer-Verlag, New York (1999).zbMATHGoogle Scholar
  21. 21.
    Mikusiński P., Sherwood H., and Taylor M. D., “Probabilistic interpretations of copulas and their convex sums,” in: Advances in Probability Distributions with Given Marginals (Rome, 1990), Kluwer Acad. Publ., Dordrecht, 1991, pp. 95–112 (Math. Appl.; V. 67).CrossRefGoogle Scholar
  22. 22.
    Heffernan P. and Resnick S., “Limit laws for random vectors with an extreme component,” Ann. Appl. Probab., 17, No. 2, 537–571 (2007).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.University of LausanneLausanneSwitzerland
  2. 2.Aarhus UniversityAarhusDenmark
  3. 3.University of LyonLyonFrance

Personalised recommendations