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. Ya. Khinchin. Another statement was given by A.N. Kolmogorov (1929). Probability theory is a branch of mathematics concerned with analysis of random phenomena. ... In mathematics, computer science, and physics, a random walk, sometimes called a drunkards walk, is a formalisation of the intuitive idea of taking successive steps, each in a random direction. ... Aleksandr Ya. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Soviet mathematician who made major advances in the fields of probability theory and topology. ...

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

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

See also: Brownian motion Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

References

• A. Khintchine. "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)