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Journal of Theoretical Probability

, Volume 28, Issue 2, pp 721–725 | Cite as

The Limit Law of the Iterated Logarithm

  • Xia Chen
Article

Abstract

For the partial sum \(\{S_n\}\) of an i.i.d. sequence with zero mean and unit variance, it is pointed out that
$$\begin{aligned} \lim _{n\rightarrow \infty }(2\log \log n)^{-1/2}\max _{1\le k\le n}{S_k\over \sqrt{k}} =1\quad \mathrm{{a.s}}. \end{aligned}$$

Keywords

The limit law of the iterated logarithm Brownian motion  Ornstein–Uhlenbeck process 

Mathematics Subject Classification (2010)

60F15 60G10 60G15 60G50 

Notes

Acknowledgments

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

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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