Skip to main content
SpringerLink
Log in

Integro-local and integral theorems for sums of random variables with semiexponential distributions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain some integro-local and integral limit theorems for the sums S(n) = ξ(1) + ⋯ + ξ(n) of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form \(P(\xi \ge t) = e^{ - t^\beta L(t)} \), where β ∈ (0, 1) and L(t) is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as x → ∞ of the probabilities P(S(n) ∈ [x, x + Δ)) and P(S(n) ≥ x) in the zone of normal deviations and all zones of large deviations of x: in the Cramér and intermediate zones, and also in the “extreme” zone where the distribution of S(n) is approximated by that of the maximal summand.

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. Borovkov A. A., “Large deviation probabilities for random walks with semiexponential distributions,” Siberian Math. J., 41, No. 6, 1061–1093 (2000).

    Article  MathSciNet  Google Scholar 

  2. Borovkov A. A. and Borovkov K. A., Asymptotic Analysis of Random Walks. Part I: Slowly Decreasing Jumps [in Russian], Nauka, Moscow (to appear).

  3. Borovkov A. A. and Mogul’skii A. A., “Integro-local limit theorems including large deviations for sums of random vectors. I,” Theory Probab. Appl., 43, 1–12 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  4. Ibragimov I. A. and Linnik Yu. V., Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing Company, Groningen (1971).

    MATH  Google Scholar 

  5. Osipov L. V., “On probabilities of large deviations for sums of independent random variables,” Theory Probab. Appl., 17, 309–331 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  6. Nagaev S. V., “Large deviations for sums of independent random variables,” Ann. Probab., 7, No. 5, 745–789 (1979).

    MATH  MathSciNet  Google Scholar 

  7. Rozovskii L. V., “Probabilities of large deviations on the whole axis,” Theory Probab. Appl., 38, No. 1, 53–79 (1993).

    Article  MathSciNet  Google Scholar 

  8. Nagaev S. V., “Some limit theorems for large deviations,” Theory Probab. Appl., 10, 214–235 (1965).

    Article  MathSciNet  Google Scholar 

  9. Petrov V. V., “Limit theorems for large deviations when the Cramér condition is violated,” Vestnik Leningrad Univ., I: No. 19, 49–68 (1963); II: No. 1, 58–75 (1968).

  10. Wolf W., “Probabilities for large deviations if the Cramér condition is not satisfied,” Math. Nachr., 70, 197–215 (1975).

    MATH  MathSciNet  Google Scholar 

  11. Wolf W., “Asymptotische Entwicklungen fuer Wahrscheinlichkeiten grosser Abweichungen,” Z. Wahrsch. Verw. Gebiete, Bd 40, 239–256 (1977).

    Article  MATH  Google Scholar 

  12. Saulis L. and Statulevicius V., Limit Theorems for Large Deviations, Kluwer Academic Publishers, Dordrecht etc. (1991).

    MATH  Google Scholar 

  13. Mikosh T. and Nagaev A. V., “Large deviations of heavy-tailed sums with applications in insurance,” Extremes, 1, No. 1, 81–110 (1998).

    Article  MathSciNet  Google Scholar 

  14. Nagaev A. V., “Integral limit theorems taking deviations into account when Cramér’s condition does not hold,” Teor. Veroyatnost. i Primenen., 14, I: No. 1, 51–64; II: No 2, 203–214 (1969).

    MATH  MathSciNet  Google Scholar 

  15. Nagaev A. V., “On a property of sums of independent random variables,” Teor. Veroyatnost. i Primenen., 22, No. 2, 335–346 (1977).

    MathSciNet  Google Scholar 

  16. Pinelis I. F., “The asymptotic equivalence of probabilities of large deviations for the sums and the maximum of independent random variables,” in: Trudy Inst. Mat. (Novosibirsk), Novosibirsk, 5, 1985, pp. 144–173.

    MathSciNet  Google Scholar 

  17. Borovkov A. A., “Estimates for the distribution of sums and maxima of sums of random variables without the Cramér condition,” Siberian Math. J., 41, No. 5, 811–848 (2000).

    MathSciNet  Google Scholar 

  18. Korshunov D. A., “Large deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution,” Theory Probab. Appl., 46, No. 2, 355–366 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  19. Aleshkyavichene A. K., “The Cramér approximation with Cramér’s condition violated,” Liet. Mat. Rink., 45, No. 4, 447–456 (2005).

    MathSciNet  Google Scholar 

  20. Aleshkyavichene A. K., “On the probabilities of large deviations for the maximum of sums of independent random variables,” Teor. Veroyatnost. i Primenen., 24, No. 1, 322–337 (1979).

    MathSciNet  Google Scholar 

  21. Stone C., “On local and ratio limit theorems,” in: Proc. of the Fifth Berkeley Sympos. on Mathematical Statistics and Probability. Vol. II, part II, Univ. California Press, Berkeley and Los Angeles, 1966, pp. 217–224.

    Google Scholar 

  22. Stone C., “A local limit theorem for nonlattice multi-dimensional distribution functions,” Ann. Math. Statist., 36, 546–551 (1965).

    MathSciNet  Google Scholar 

  23. Borovkov A. A. and Mogul’skii A. A., “Integro-local theorems for sums of random vectors in a series scheme,” Mat. Zametki, 79, No. 4, 468–482 (2006).

    MathSciNet  Google Scholar 

  24. Rozovskii L. V., “Probabilities of large deviations of sums of independent random variables with a common distribution function that belongs to the domain of attraction of a normal law,” Theory Probab. Appl., 34, No. 4, 625–644 (1989).

    Article  MathSciNet  Google Scholar 

  25. Pinelis I. F., “A problem on large deviations in a space of trajectories,” Theory Probab. Appl., 26, 69–84 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  26. Shepp L. A., “A local limit theorem,” Ann. Math. Statist., 35, 419–423 (1964).

    MathSciNet  Google Scholar 

  27. Borovkov A. A., “On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums,” Siberian Math. J., 43, No. 6, 995–1022 (2002).

    Article  MathSciNet  Google Scholar 

  28. Borovkov A. A. and Mogul’skii A. A., “Large deviations and super large deviations of the sums of independent random variables with the Cramér condition. I,” Teor. Veroyatnost. i Primenen., 51, No. 2, 260–294 (2006).

    Google Scholar 

  29. Borovkov A. A. and Mogul’skii A. A., “Large deviations and super large deviations of the sums of independent random variables with the Cramér condition. II,” Teor. Veroyatnost. i Primenen., 51, No. 4 (2006).

  30. Gnedenko B. D., “On a local limit theorem of probability,” Uspekhi Mat. Nauk, 3, No. 3, 187–194 (1948).

    MathSciNet  Google Scholar 

  31. Gnedenko B. V. and Kolmogorov A. N., Limit Distributions for Sums of Independent Random Variables [in Russian], GITTL, Moscow; Leningrad (1949).

    Google Scholar 

  32. Rvacheva E. L., “On domains of attraction of multidimensional distributions,” Uchenye Zapiski L’vov Univ. Ser. Mekh.-Mat., 6, 5–44 (1958).

    Google Scholar 

  33. Borovkov A. A. and Mogul’skii A. A., Large Deviations and Testing of Statistical Hypotheses [in Russian], Nauka, Novosibirsk (1992).

    Google Scholar 

  34. Demidovich B. P., Problems and Exercises in Mathematical Analysis [in Russian], Nauka, Moscow (1972).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. A. Borovkov & A. A. Mogul’skii

Authors
  1. A. A. Borovkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Mogul’skii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text Copyright © 2006 Borovkov A. A. and Mogul’skiĭ A. A.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 6, pp. 1218–1257, November–December, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borovkov, A.A., Mogul’skii, A.A. Integro-local and integral theorems for sums of random variables with semiexponential distributions. Sib Math J 47, 990–1026 (2006). https://doi.org/10.1007/s11202-006-0110-x

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-006-0110-x

Keywords

Search

Navigation