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

• ORIGINAL ARTICLES
• Published:

## Abstract

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.

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

## References

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)

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

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

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)

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

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

## Author information

Authors

### Corresponding author

Correspondence to Ye Jiang.

Research supported by National Nature Science Foundation of China: 10471126

## Rights and permissions

Reprints and permissions

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). https://doi.org/10.1007/s10114-005-0615-4

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s10114-005-0615-4