Journal of Theoretical Probability

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

The Limit Law of the Iterated Logarithm

  • Xia Chen


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


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

Mathematics Subject Classification (2010)

60F15 60G10 60G15 60G50 



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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