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).

<math> \limsup_{n \to \infty} \frac{|S_n|}{\sigma\sqrt{2 n \log \log n}} = 1

\quad \mbox{(almost surely)} </math>

where S_n is the sum of n independent, identically distributed variables with mean zero and finite variance σ².

See also: Brownian motion

References

• A. Khinchine. "Über einen Satz die Wahrscheinlichkeitsrechnung", Fundamenta Mathematica, 6:9-20, 1924. (The author's name is shown here in an alternate transliteration.)
• Leo Breiman. Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. (See Sections 3.9, 12.9, and 12.10.)
• A. Kolmogoroff. "Über das Gesetz des iterierten Logarithmus". Mathematische Annalen, 101:126-135, 1929. (At the Göttinger DigitalisierungsZentrum web site)