Plot of (red), its standard deviation (blue) and its bound given by LIL (green). Notice the way it randomly switches from the upper bound to the lower bound. Both axes are non-linearly transformed (as explained in figure summary) to make this effect more visible.
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924).[1] Another statement was given by A. N. Kolmogorov in 1929.[2]
^A. Khinchine. "Über einen Satz der Wahrscheinlichkeitsrechnung", Fundamenta Mathematicae 6 (1924): pp. 9–20 (The author's name is shown here in an alternate transliteration.)
^A. Kolmogoroff. "Über das Gesetz des iterierten Logarithmus". Mathematische Annalen, 101: 126–135, 1929. (At the Göttinger DigitalisierungsZentrum web site)
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