重对数律

概率论中,重对数律(LIL)用来描述一个随机游走的振幅。其最早为Aleksandr Y. Khinchin在1924年所叙述[1];之后Andrey N. Kolmogorov在1929年给出了另一个叙述[2]。由于定理中出现了二重对数,故名。

内容

是一列独立同分布的随机变量,其期望为0,方差为1;且记,那么:

其中“log”是自然对数,“lim sup”是上极限,“a.s.”是“几乎必然[3]

参见

中心极限定理

参考文献

  1. A. Khinchine. "Über einen Satz der Wahrscheinlichkeitsrechnung", Fundamenta Mathematica, 6:9-20, 1924. (The author's name is shown here in an alternate transliteration.)
  2. A. Kolmogoroff. "Über das Gesetz des iterierten Logarithmus" 页面存档备份,存于. Mathematische Annalen, 101:126-135, 1929. (At the Göttinger DigitalisierungsZentrum web site 页面存档备份,存于)
  3. 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.)
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