Skip to main content
SpringerLink
Log in

Asymptotic property for some series of probability

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

Let {X, X n , n ≥ 1} be a sequence of i.i.d.random variables with zero mean, and set \(\sum\limits_{k = 1}^n {X_k }\), EX 2 = σ 2 > 0, \(\lambda \left( \varepsilon \right) = \sum\limits_{n = 1}^\infty {P\left( {\left| {S_n } \right| \geqslant n\varepsilon } \right)}\). In this paper, we discuss the rate of the approximation of σ 2 by ɛ 2 λ(ɛ) under suitable conditions, and improve the corresponding results of Klesov (1994).

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. Chen, R. A remark on the tail probability of a distribution. J. Multivariate Anal., 8: 328–333 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Erdös, P. On a theorem of Hsu and Robbins. Ann. Math. Statist., 20: 286–291 (1949)

    Article  MATH  Google Scholar 

  3. Erdös, P. Remark on my paper “On a theorem of Hsu and Robbins”. Ann. Math. Statist., 21: 138–138 (1950)

    Article  MATH  Google Scholar 

  4. Gut, A., Spǧtaru, A. Precise asymptotics in the Baum-Katz and Davis law of large numbers. J. Math. Anal. Appl., 248: 233–246 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gut, A., Spǧtaru, A. Precise asymptotics in the law of the iterated logarithm. Ann. Probab., 28: 1870–1883 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gut, A., Spǧtaru, A. Precise asymptotics in some strong limit theorem for multidimensionally indexed random variables. J. Multivariate Anal., 86: 398–422 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Heyde, C.C. A supplement to the strong law of large numbers. J. Appl. Probab., 12: 903–907 (1975)

    Article  MathSciNet  Google Scholar 

  8. Hsu, P.L., Robbins, H. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A Math., 33: 25–31 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, W., Zhang, L.X. Precise asymptotics in the law of logarithm in the Hilbert space. J. Math. Anal. Appl., 304: 734–758 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Klesov, O.I. On the convergence rate in a theorem of Heyde. Theor. Probability and Math. Statist., 49: 83–87 (1994)

    MathSciNet  Google Scholar 

  11. Lanzinger, H., Stadtmüller, U. Refined Baum-Katz laws for weighted sums of i.i.d. random variables. Statist. Probab. Lett., 69: 357–368 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, W.D., Lin, Z.Y. Precise asymptotics for a new kind of complete moment convergence. Statist. Probab. Lett., 76: 1787–1799 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nagaev, S.V. Some limit theorems for large deviation. Theory Probab. Appl., 10: 214–235 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Osipov, L.V., Petrov, V.V. On an estimate of the remainder term in the central limit theorem. Teor. Veroyatn. Primen., 12: 322–329 (1967)

    MathSciNet  MATH  Google Scholar 

  15. Spǧtaru, A. Exact asymptotics in log log laws for random fields. J. Theoret. Probab., 17: 943–965 (2004)

    Article  MathSciNet  Google Scholar 

  16. Spǧtaru, A. Precise asymptotics for a series of T.L. Lai. Proc. Amer. Math. Soc., 132: 3387–3395 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, China Jiliang University, Hangzhou, 310018, China

    Jian-jun He & Ting-fan Xie

Authors
  1. Jian-jun He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting-fan Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-jun He.

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, Jj., Xie, Tf. Asymptotic property for some series of probability. Acta Math. Appl. Sin. Engl. Ser. 29, 179–186 (2013). https://doi.org/10.1007/s10255-012-0138-6

Download citation

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-012-0138-6

Keywords

2000 MR Subject Classification

Search

Navigation