Law of the iterated logarithm

In probability theory, the law of the iterated logarithm is the name given to several theorems which describe the magnitude of the fluctuations of a random walk. The original statement (1924) of the law of the iterated logarithm is due to A. Y. Khinchin. Another statement was given by A.N. Kolmogorov (1929).

One of the simpler forms of the law of the iterated logarithm can be stated as follows (Theorem 3.52 in Breiman).


 * $$ \limsup_{n \to \infty} \frac{|S_n|}{\sigma\sqrt{2 n \log \log n}} = 1

\quad \mbox{(almost surely)} $$

where Sn is the sum of n independent, identically distributed variables with mean zero and finite variance &sigma;2.

See also: Brownian motion