Skip to main content
Log in

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript


Let {X,X n ; n ≥ 1} be a sequence of i.i.d. random variables, EX = 0, EX 2 = σ 2 < ∞. Set S n = X 1 + X 2 + ⋯ + X n , M n = max kn S k ∣, n ≥ 1. Let a n = O(1/ log log n). In this paper, we prove that, for b > −1,

$$ \begin{aligned} & {\mathop {\lim }\limits_{\varepsilon \searrow 0} }\varepsilon ^{{2{\left( {b + 1} \right)}}} {\sum\limits_{n = 1}^\infty {\frac{{{\left( {\log \;\log \;n} \right)}^{b} }} {{n\log n}}} }n^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\rm E}{\left\{ {M_{n} - \sigma {\left( {\varepsilon + a_{n} } \right)}{\sqrt {2n\log \;\log \;n} }} \right\}} + \\ & = \frac{{\sigma 2^{{ - b}} }} {{{\left( {b + 1} \right)}{\left( {2b + 3} \right)}}}{\rm E}{\left| N \right|}^{{2b + 3}} {\sum\limits_{k = 0}^\infty {\frac{{{\left( { - 1} \right)}^{k} }} {{{\left( {2k + 1} \right)}^{{2b + 3}} }}} } \\ \end{aligned} $$

holds if and only if EX = 0 and EX 2 = σ 2 < ∞.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Li, D., Wang, X. C., Rao, M. B.: Some results on convergence rates for probabilities of moderate deviations for sums of random variables. Internet. J. Math and Math. Sci., 15(3), 481–498 (1992)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math Academia Sinica, 16, 177–201 (1988)

    MATH  Google Scholar 

  4. Wang, D. C., Su, chun.: Moment complete convergence for B valued I.I.D. random variables. Acta Mathematicae Applicatae Sinica (in Chinese), in press

  5. Csörgö, M., Révész, P.: Strong Approximations in Probability and Statistics, Academic Press, New York, 1981

  6. Sakhanenko, A. I.: On unimprovable estimates of the rate of convergence in the invariance principle. In Colloquia Math. Soci. János Bolyai, 32, 779–783 (1980), Nonparametric Statistical Inference, Budapest (Hungary)

  7. Sakhanenko, A. I.: On estimates of the rate of convergence in the invariance principle. In Advances in Probab. Theory: Limit Theorems and Related Problems (A. A. Borovkov, Ed.), Springer, New York, 124–135, 1984

  8. Sakhanenko, A. I.: Convergence rate in the invariance principle for nonidentically distributed variables with exponential moments. In Advances in Probab. Theory: Limit Theorems for Sums of Random Variables (A. A. Borovkor, Ed.), Springer, New York, 2–73, 1985

  9. Billingsley, P.: Convergence of Probability Measures, J. Wiley, New York, 1968

  10. Einmahl, U.: The Darling–Erdő Theorem for sums of i.i.d. random variables. Probab. Theory Relat. Fields, 82, 241–257 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Feller, W.: The law of the iterated logarithm for idnetically distributed random variables. Ann. Math., 47, 631–638 (1945)

    Article  MathSciNet  Google Scholar 

  12. Petrov, V. V.: Limit Theorem of Probability Theory, Oxford Univ. Press, Oxford, 1995

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ye Jiang.

Additional information

Research supported by National Nature Science Foundation of China: 10471126

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, Y., Zhang, L.X. Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables. Acta Math Sinica 22, 781–792 (2006).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


MR (2000) Subject Classification