Methodology and Computing in Applied Probability

, Volume 13, Issue 4, pp 821–833 | Cite as

Precise Large Deviations of Random Sums in Presence of Negative Dependence and Consistent Variation

Article

Abstract

The study of precise large deviations for random sums is an important topic in insurance and finance. In this paper, we extend recent results of Tang (Electron J Probab 11(4):107–120, 2006) and Liu (Stat Probab Lett 79(9):1290–1298, 2009) to random sums in various situations. In particular, we establish a precise large deviation result for a nonstandard renewal risk model in which innovations, modelled as real-valued random variables, are negatively dependent with common consistently-varying-tailed distribution, and their inter-arrival times are also negatively dependent.

Keywords

Consistent variation Counting process Lower/upper extended negative dependence Precise large deviation Uniformity 

AMS 2000 Subject Classifications

60F10 60E15 62H20 62E20 

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References

  1. Baltrūnas A, Leipus R, Šiaulys J (2008) Precise large deviation results for the total claim amount under subexponential claim sizes. Stat Probab Lett 78(10):1206–1214MATHCrossRefGoogle Scholar
  2. Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge University Press, CambridgeMATHGoogle Scholar
  3. Block HW, Savits TH, Shaked M (1982) Some concepts of negative dependence. Ann Probab 10(3):765–772MathSciNetMATHCrossRefGoogle Scholar
  4. Cline DBH, Samorodnitsky G (1994) Subexponentiality of the product of independent random variables. Stoch Process Their Appl 49(1):75–98MathSciNetMATHCrossRefGoogle Scholar
  5. Ebrahimi N, Ghosh M (1981) Multivariate negative dependence. Commun Stat A Theory Methods 10(4):307–337MathSciNetCrossRefGoogle Scholar
  6. Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, BerlinMATHGoogle Scholar
  7. Kaas R, Tang Q (2005) A large deviation result for aggregate claims with dependent claim occurrences. Insur, Math Econ 36(3):251–259MathSciNetMATHCrossRefGoogle Scholar
  8. Klüppelberg C, Mikosch T (1997) Large deviations of heavy-tailed random sums with applications in insurance and finance. J Appl Probab 34(2):293–308MathSciNetMATHCrossRefGoogle Scholar
  9. Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37(5):1137–1153MathSciNetMATHCrossRefGoogle Scholar
  10. Lin J (2008) The general principle for precise large deviations of heavy-tailed random sums. Stat Probab Lett 78(6):749–758MATHCrossRefGoogle Scholar
  11. Liu Y (2007) Precise large deviations for negatively associated random variables with consistently varying tails. Stat Probab Lett 77(2):181–189MATHCrossRefGoogle Scholar
  12. Liu L (2009) Precise large deviations for dependent random variables with heavy tails. Stat Probab Lett 79(9):1290–1298MATHCrossRefGoogle Scholar
  13. Liu Y, Hu Y (2003) Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails. Sci China, Ser A 46(3):383–395MathSciNetMATHGoogle Scholar
  14. Matuła P (1992) A note on the almost sure convergence of sums of negatively dependent random variables. Stat Probab Lett 15(3):209–213MATHCrossRefGoogle Scholar
  15. McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Concepts, techniques and tools. Princeton University Press, PrincetonMATHGoogle Scholar
  16. Mikosch T, Nagaev AV (1998) Large deviations of heavy-tailed sums with applications in insurance. Extremes 1(1):81–110MathSciNetMATHCrossRefGoogle Scholar
  17. Ng KW, Tang Q, Yan J, Yang H (2003) Precise large deviations for the prospective-loss process. J Appl Probab 40(2):391–400MathSciNetMATHCrossRefGoogle Scholar
  18. Ng KW, Tang Q, Yan J, Yang H (2004) Precise large deviations for sums of random variables with consistently varying tails. J Appl Probab 41(1):93–107MathSciNetMATHCrossRefGoogle Scholar
  19. Shen X, Lin Z (2008) Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails. Stat Probab Lett 78(18):3222–3229MathSciNetMATHCrossRefGoogle Scholar
  20. Tang Q (2006) Insensitivity to negative dependence of the asymptotic behavior of precise large deviations. Electron J Probab 11(4):107–120MathSciNetGoogle Scholar
  21. Tang Q, Tsitsiashvili G (2003) Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch Process Their Appl 108(2):299–325MathSciNetMATHGoogle Scholar
  22. Tang Q, Su C, Jiang T, Zhang J (2001) Large deviations for heavy-tailed random sums in compound renewal model. Stat Probab Lett 52(1):91–100MathSciNetMATHCrossRefGoogle Scholar
  23. Wang S, Wang W (2007) Precise large deviations for sums of random variables with consistently varying tails in multi-risk models. J Appl Probab 44(4):889–900MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesThe University of LiverpoolLiverpoolUK
  2. 2.Department of Statistics and Actuarial ScienceThe University of Hong KongHong KongHong Kong

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