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The Limit Law of the Iterated Logarithm

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Abstract

For the partial sum \(\{S_n\}\) of an i.i.d. sequence with zero mean and unit variance, it is pointed out that

$$\begin{aligned} \lim _{n\rightarrow \infty }(2\log \log n)^{-1/2}\max _{1\le k\le n}{S_k\over \sqrt{k}} =1\quad \mathrm{{a.s}}. \end{aligned}$$

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Acknowledgments

Research of Xia Chen was partially supported by the Simons Foundation #244767.

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  1. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Xia Chen

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  1. Xia Chen
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Correspondence to Xia Chen.

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Chen, X. The Limit Law of the Iterated Logarithm. J Theor Probab 28, 721–725 (2015). https://doi.org/10.1007/s10959-013-0481-4

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  • DOI: https://doi.org/10.1007/s10959-013-0481-4

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