Skip to main content
SpringerLink
Log in

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Let ξ1,ξ2,... be independent random variables with distributions F 1 F 2,... in a triangular array scheme (F i may depend on some parameter). Assume that Eξ i = 0, Eξ 2i < ∞, and put \(S_n = \sum {_{i = 1}^n \;} \xi _i ,\;\overline S _n = \max _{k \leqslant n} S_k\). Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution \(F = \frac{1}{n}\sum {_{i = 1}^n F_i }\), we find upper and lower bounds for the probabilities P(S n > x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {S k } crosses the remote boundary {g(k)}; that is, the asymptotics of P(max k≤n (S k − g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  4. Borovkov A. A., “Notes on inequalities for sums of independent variables,” Theory Probab. Appl., 17, No.3, 556–557 (1972).

    MATH  Google Scholar 

  5. Borovkov A. A. and Borovkov K. A., “On probabilities of large deviations for random walks. I. Regularly varying distribution tails,” Theory Probab. Appl., 46, No.2, 193–213 (2001).

    MathSciNet  Google Scholar 

  6. Borovkov A. A., “Boundary value problems, the invariance principle, and large deviations,” Uspekhi Mat. Nauk, 38, No.4, 227–254 (1983).

    MATH  MathSciNet  Google Scholar 

  7. Borovkov A. A., Mogul'skii A. A., and Sakhanenko A. I., “Limit theorems for random processes,” in: Probability Theory 7 [in Russian] (Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya; Vol. 82), VINITI, Moscow, 1995, pp. 5–199.

    Google Scholar 

  8. Borovkov A. A., “Limit theorems on the distributions of maxima of sums of bounded lattice random variables. I, II,” Theory Probab. Appl., 5, No.2, 137–171 (1960); 5, No. 4, 377–392 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  9. Borovkov A. A., “New limit theorems in boundary value problems for sums of independent summands,” Siberian Math. J., 3, No.5, 645–694 (1962).

    MATH  MathSciNet  Google Scholar 

  10. Borovkov A. A., “On the limit conditional distributions connected with large deviations,” Siberian Math. J., 37, No.4, 635–646 (1996).

    MATH  MathSciNet  Google Scholar 

  11. Fuk Dao Ha and Nagaev S. V., “Probability inequalities for sums of independent random variables,” Theory Probab. Appl., 16, No.4, 643–660 (1971).

    Google Scholar 

  12. Nagaev S. V., “Probabilities of large deviations in Banach space,” Mat. Zametki, 34, No.2, 309–313 (1983).

    MathSciNet  Google Scholar 

  13. Borovkov A. A., “Large deviations for random walks with nonidentically distributed jumps having infinite variance,” Siberian Math. J., 46, No.1, 35–55 (2005).

    Article  MathSciNet  Google Scholar 

  14. Borovkov A. A., Stochastic Processes in Queueing Theory [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  15. Prokhorov Yu. V., “Transient phenomena in queueing processes,” Litovsk. Mat. Sb., 3, No.1, 199–206 (1963).

    MATH  MathSciNet  Google Scholar 

  16. Kingman F. G., “On queues in heavy traffic,” J. R. Statist. Soc. Ser. B, 24, 383–392 (1962).

    MATH  MathSciNet  Google Scholar 

  17. Feller W., An Introduction to Probability Theory and Its Applications. Vol. 2., John Wiley and Sons, Inc., New York etc. (1971).

    Google Scholar 

  18. Billingsley P., Convergence of Probability Measures, Wiley, Chichester (1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. A. Borovkov

Authors
  1. A. A. Borovkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text Copyright © 2005 Borovkov A. A.

The author was supported by the Russian Foundation for Basic Research (Grant 02-01-00902) and the Grant Council of the President of the Russian Federation (Grant NSh-2139.2003.1).

__________

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 1265–1287, November–December, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borovkov, A.A. Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance. Sib Math J 46, 1020–1038 (2005). https://doi.org/10.1007/s11202-005-0097-8

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-005-0097-8

Keywords

Search

Navigation