Skip to main content
SpringerLink
Log in

Precise Large Deviations for Long-Tailed Distributions

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

In this paper, we investigate the precise large deviations for sums of independent identically distributed random variables with heavy-tailed distributions. We prove asymptotic relations for non-random sums and for random sums of random variables with long-tailed distributions. We apply the results on two useful counting processes, namely, renewal and compound-renewal processes.

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. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  2. Cline, D.B.H., Hsing, T.: Large deviation probabilities for sums and maxima of r.v.’s with heavy or subexponential tails. Preprint, Texas AM University (1991)

  3. Cline, D.B.H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Appl. 49(1), 75–98 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Čistjakov, V.P.: A theorem on sums of independent positive random variables and its applications to branching random processes. Teor. Veroatn. Primen. 9, 710–718 (1964)

    Google Scholar 

  5. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events. Springer, Berlin (1997)

    MATH  Google Scholar 

  6. Foss, S., Korshunov, D., Zachary, S.: Convolutions of long-tailed and subexponential distributions. J. Appl. Probab. 46(3), 756–767 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Foss, S., Korshunov, D., Zachary, S.: A introduction to heavy-tailed and subexponential distributions. Oberwolfach Preprints (2009)

  8. Gut, A.: Stopped Random Walk. Limit Theorem and Applications. Springer, New York (1998)

    Google Scholar 

  9. Heyde, C.C.: A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 7, 303–308 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Kaas, R., Tang, Q.: A large deviation result for the aggregate claims with dependent claim occurrences. Insur. Math. Econ. 36, 251–259 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Konstantinides, D., Loukissas, F.: Precise large deviations for consistently varying-tailed distributions in the compound renewal risk model. Lith. Math. J. 50(4), 391–400 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Nagaev, A.V.: Integral limit theorems with regard to large deviations when Cramér’s condition is not satisfied. I. Teor. Veroatn. Primen. 14, 51–63 (1969)

    MathSciNet  Google Scholar 

  17. Nagaev, A.V.: Integral limit theorems for large deviations when Cramer’s condition is not fulfilled II. Theory Probab. Appl. 14, 193–208 (1969)

    Article  MATH  Google Scholar 

  18. Nagaev, S.V.: Large deviations for sums of independent random variables. In: Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Tech. Univ., Prague, pp. 657–674. Academia, Prague. Dedicated to the memory of Antonín Špaček (1971)

    Google Scholar 

  19. Nagaev, S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7(5), 745–789 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ng, K.W., Tang, Q., Yan, J., Yang, H.: Precise large deviations for sums of random variables with consistently varying tails. J. Appl. Probab. 41(1), 93–107 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Paulauskas, V., Skučaitė, A.: Some asymptotic results for one-sided large deviation probabilities. Lith. Math. J. 43(3), 318–326 (2003)

    Article  MATH  Google Scholar 

  22. Pinelis, I.F.: On the Asymptotic Equivalence of Probabilities of Large Deviation for Sums and Maxima of Independent Random Variables. Trudy Inst. Math., vol. 5. Nauka, Novosibirsk (1985)

    Google Scholar 

  23. Skučaitė, A.: Large deviations for sums of independent heavy-tailed random variables. Lith. Math. J. 44(2), 198–208 (2004)

    Article  MATH  Google Scholar 

  24. Tang, Q., Su, C., Jiang, T., Zang, J.S.: Large deviation for heavy-tailed random sums in compound renewal model. Stat. Probab. Lett. 44, 91–100 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi, 83 200, Samos, Greece

    Fotis Loukissas

Authors
  1. Fotis Loukissas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fotis Loukissas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Loukissas, F. Precise Large Deviations for Long-Tailed Distributions. J Theor Probab 25, 913–924 (2012). https://doi.org/10.1007/s10959-011-0367-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-011-0367-2

Keywords

Mathematics Subject Classification (2010)

Search

Navigation