Skip to main content
SpringerLink
Log in

Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

This paper studies the moderate deviations of real-valued extended negatively dependent (END) random variables with consistently varying tails. The moderate deviations of partial sums are first given. The results are then used to establish the necessary and sufficient conditions for the moderate deviations of random sums under certain circumstances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Block H W, Savits T H, Shaked M. Some concepts of negative dependence. Ann Probab, 1982, 10: 765–772

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen Y, Zhang W P. Large deviation for random sums of negatively dependent random variables with consistently varying tails. Statist Probab Lett, 2007, 77: 530–538

    Article  MATH  MathSciNet  Google Scholar 

  3. Cline D B H, Hsing T. Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint, Texas A&M University, 1991

  4. Cline D B H, Samorodnitsky G. Subexponentiality of the product of independent random variables. Stochastic Process Appl, 1994, 49: 75–98

    Article  MATH  MathSciNet  Google Scholar 

  5. Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. Berlin: Springer, 1997

    MATH  Google Scholar 

  6. Gao F Q. Moderate deviations for random sums of heavy-tailed random variables. Acta Math Sin (Engl Ser), 2007, 23: 1527–1536

    Article  MATH  MathSciNet  Google Scholar 

  7. Heyde C C. On large deviation problems for sums of random variables which are not attracted to the normal law. Ann Math Statist, 1967, 38: 1575–1578

    Article  MATH  MathSciNet  Google Scholar 

  8. Heyde C C. On large deviation probabilities in the case of attraction to a non-normal stable law. Sankhyā Ser A, 1968, 30: 253–258

    MATH  MathSciNet  Google Scholar 

  9. Jelenković P R, Lazar A A. Asymptotic results for multiplexing subexponential on-off processes. Adv Appl Probab, 1999, 31: 394–421

    Article  MATH  Google Scholar 

  10. Klüppelberg C, Mikosch T. Large deviations of heavy-tailed random sums with applications in insurance and finance. J Appl Probab, 1997, 34: 293–308

    Article  MATH  MathSciNet  Google Scholar 

  11. Lehmann E L. Some concepts of dependence. Ann Math Statist, 1966, 37: 1137–1153

    Article  MATH  MathSciNet  Google Scholar 

  12. Lin J. The general principle for precise large deviations of heavy-tailed random sums. Statist Probab Lett, 2008, 78: 749–758

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu L. Precise large deviations for dependent random variables with heavy tails. Statist Probab Lett, 2009, 79: 1290–1298

    Article  MATH  MathSciNet  Google Scholar 

  14. Meerschaert M M, Scheffler H P. Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. New York: Wiley, 2001

    Google Scholar 

  15. Mikosch T, Nagaev A V. Large deviations of heavy-tailed sums with applications in insurance. Extremes, 1998, 1: 81–110

    Article  MATH  MathSciNet  Google Scholar 

  16. Nagaev A V. Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I,II. Theory Probab Appl, 1969, 14: 51–64 and 193–208

    Article  MathSciNet  Google Scholar 

  17. Nagaev S V. Large deviations of sums of independent random variables. Ann Probab, 1979, 7: 745–789

    Article  MATH  MathSciNet  Google Scholar 

  18. Ng K W, Tang Q, Yan J, et al. Precise large deviations for sums of random variables with consistently varying tails. J Appl Probab, 2004, 41: 93–107

    Article  MATH  MathSciNet  Google Scholar 

  19. Rozovski L V. Probabilities of large deviations on the whole axis. Theory Probab Appl, 1993, 38: 53–79

    Article  MathSciNet  Google Scholar 

  20. Schlegel S. Ruin probabilities in perturbed risk models. Insurance Math Econom, 1998, 22: 93–104

    Article  MATH  MathSciNet  Google Scholar 

  21. Tang Q. Insensitivity to negative dependence of the asymptotic behavior of precise large deviations. Electron J Probab, 2006, 11: 107–120

    MathSciNet  Google Scholar 

  22. Tang Q, Su C, Jiang T, et al. Large deviations for heavy-tailed random sums in compound renewal model. Statist Probab Lett, 2001, 52: 91–100

    Article  MATH  MathSciNet  Google Scholar 

  23. Tang Q, Tsitsiashvili G. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process Appl, 2003, 108: 299–325

    MATH  MathSciNet  Google Scholar 

  24. Wang D, Tang Q. Maxima of sums and random sums for negatively associated random variables with heavy tails. Statist Probab Lett, 2004, 68: 287–295

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Li Liu

Authors
  1. Li Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, L. Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Math. 53, 1421–1434 (2010). https://doi.org/10.1007/s11425-010-4012-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-4012-9

Keywords

MSC(2000)

Search

Navigation