Skip to main content

Small deviations of series of independent positive random variables with weights close to exponential

Abstract

Let ξ, ξ0, ξ1, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ j=0 a(j j was studied under different assumptions on the rate of decrease of the probability ℙ(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ℙ(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ℙ(ξ < x) ∼ bx α as x → 0, b > 0, a > 0, the asymptotics ℙ(S < x) is obtained in an explicit form up to the factor x o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].

This is a preview of subscription content, access via your institution.

References

  1. 1.

    A. A. Borovkov and P. S. Ruzankin, “On small deviations of series of weighted random variables,” J. Theoret. Probab. 21(3), 628–649 (2008).

    MATH  Article  Google Scholar 

  2. 2.

    R. Davis and S. Resnick, “Extremes of moving averages of random variables with finite endpoint,” Ann. Probab. 19(1), 312–328 (1991).

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    T. Dunker, M. A. Lifshits and W. Linde, “Small deviations of sums of independent variables,” Progr. Probab. 43, 59–74 (Basel: Birkhäuser, 1998); High Dimensional Probability (Oberwolfach, 1996).

    Google Scholar 

  4. 4.

    M. A. Lifshits, “On the lower tail probabilities of some random series,” Ann. Probab. 25(1), 424–442 (1997).

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    A. D. Myshkis, Linear delayed differential equations (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  6. 6.

    L. V. Rozovsky, “On probabilities of small deviations of some random sums,” in Abstracts of XIIth Russian colloquium school on statistical methods (Sochi, October, 1–7, 2005); Obozr. Prikl. Prom. Mat. 12(4), 865–866 (2005) [in Russian].

    Google Scholar 

  7. 7.

    G. N. Sytaya, “On some asymptotic representation of the Gaussian measure in a Hilbert space,” Theor. Stoch. Processes 2, 94–104 (1974).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. A. Borovkov.

Additional information

Original Russian Text © A. A. Borovkov and P. S. Ruzankin, 2008, published in Matematicheskie Trudy, 2008, Vol. 11, No. 1, pp. 49–67.

About this article

Cite this article

Borovkov, A.A., Ruzankin, P.S. Small deviations of series of independent positive random variables with weights close to exponential. Sib. Adv. Math. 18, 163–175 (2008). https://doi.org/10.3103/S1055134408030024

Download citation

Keywords

  • small deviations
  • series of independent random variables
  • delayed differential equations