Law of the iterated logarithm

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). Another statement was given by A.N. Kolmogorov in 1929.

Let {Y_n} be independent, identically distributed random variables with means zero and unit variances. Let S_n = Y₁ + … + Y_n. Then

where “log” is the natural logarithm, “lim sup” denotes the limit superior, and “a.s.” stands for “almost surely”.

The law of iterated logarithms operates “in between” the law of large numbers and the central limit theorem. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums S_n, scaled by n⁻¹, converge to zero, respectively in probability and almost surely:

On the other hand, the central limit theorem states that the sums S_n scaled by the factor n^−½ converge in distribution to a standard normal distribution. By Kolmogorov's zero–one law, for any fixed M, the probability that the event ${\displaystyle \limsup _{n}{\frac {S_{n}}{\sqrt {n}}}>M}$ occurs is 0 or 1. Then

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