## Abstract

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

holds if and only if E*X* = 0 and E*X*
^{2} = *σ*
^{2} < ∞.

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## References

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)Gut, A., Spătaru, A.: Precise asymptotics in the law of the iterated logarithm.

*Ann. Probab.*,**28**, 1870–1883 (2000)Chow, Y. S.: On the rate of moment convergence of sample sums and extremes.

*Bull. Inst. Math Academia Sinica*,**16**, 177–201 (1988)Wang, D. C., Su, chun.: Moment complete convergence for B valued I.I.D. random variables.

*Acta Mathematicae Applicatae Sinica*(in Chinese), in pressCsörgö, M., Révész, P.: Strong Approximations in Probability and Statistics, Academic Press, New York, 1981

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)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

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

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

Einmahl, U.: The Darling–Erdő Theorem for sums of i.i.d. random variables.

*Probab. Theory Relat. Fields*,**82**, 241–257 (1989)Feller, W.: The law of the iterated logarithm for idnetically distributed random variables.

*Ann. Math.*,**47**, 631–638 (1945)Petrov, V. V.: Limit Theorem of Probability Theory, Oxford Univ. Press, Oxford, 1995

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Research supported by National Nature Science Foundation of China: 10471126

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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

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