View source for 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 [[Aleksandr Khinchin|A. Y. Khinchin]].
Another statement was given by [[Andrey Kolmogorov|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''<sub>''n''</sub> is the sum of ''n'' independent, identically distributed variables with mean zero and finite variance &sigma;<sup>2</sup>.

''See also:'' [[Brownian motion]]

== References ==
* [[Aleksandr Khinchin|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.)''
* [[Andrey Kolmogorov|A. Kolmogoroff]]. [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0101 "Über das Gesetz des iterierten Logarithmus"]. ''Mathematische Annalen'', 101:126-135, 1929. ''(At the [http://gdz.sub.uni-goettingen.de/en/index.html Göttinger DigitalisierungsZentrum web site])''

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[[Category:Probability theory]]
[[Category:Stochastic processes]]
[[Category:Statistical theorems]]

[[pl:Prawo iterowanego logarytmu]]
[[de:Gesetz_des_iterierten_Logarithmus]]
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