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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. Y. Khinchin (1924).〔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.)''〕 Another statement was given by A.N. Kolmogorov in 1929.〔A. Kolmogoroff. ("Über das Gesetz des iterierten Logarithmus" ). ''Mathematische Annalen'', 101: 126-135, 1929. ''(At the (Göttinger DigitalisierungsZentrum web site ))''〕 ==Statement== Let be independent, identically distributed random variables with means zero and unit variances. Let ''S''''n'' = ''Y''1 + … + ''Y''''n''. Then : where “log” is the natural logarithm, “lim sup” denotes the limit superior, and “a.s.” stands for “almost surely”.〔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; Theorem 3.52 specifically.)''〕 〔Varadhan, S. R. S. Stochastic processes. Courant Lecture Notes in Mathematics, 16. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2007.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Law of the iterated logarithm」の詳細全文を読む スポンサード リンク
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