Journal of Theoretical Probability

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

The Limit Law of the Iterated Logarithm



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.


  1. 1.
    Bingham, N.H.: Variants on the law of the iterated logarithm. Bull. Lond. Math. Soc. 18, 433–467 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chung, K.L.: On the maximum partial sums of sequences of independent random variables. Trans. Am. Math. Soc. 64, 205–233 (1948)CrossRefGoogle Scholar
  3. 3.
    de Acosta, A.: A new proof of the Hartman–Wintner law of the iterated logarithm. Ann. Probab. 11, 270–276 (1983)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hartman, P., Wintner, A.: On the law of the iterated logarithm. Am. J. Math. 63, 169–176 (1941)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jain, N.C., Pruitt, W.E.: The other law of the iterated logarithm. Ann. Probab. 3, 1046–1049 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Marcus, M.B., Rosen, J.: Markov processes, Gaussian processes, and local times. Cambridge studies in advanced mathematics, p. 100. Cambridge University Press, New York (2006)CrossRefGoogle Scholar
  7. 7.
    Mittal, Y.: Limiting behavior of maxima in stationary Gaussian sequences. Ann. Probab. 2, 231–242 (1974)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Pickands, J.: An iterated logarithm law for the maximum of a stationary Gaussian sequence. Z. Wahrsch. Verw. Gebiete 12, 344–353 (1969)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Slepian, D.: The one-sided barrier problem for Gaussian noise. Bell Syst. Tech. J. 41, 463–501 (1962)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 49, 23–32 (1964)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

Personalised recommendations