Loi du logarithme itéré

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). Another statement was given by A. N. Kolmogorov in 1929. Let {Yn} be independent, identically distributed random variables with means zero and unit variances. Let Sn = Y1 + ... + Yn. 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 Sn, 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 Sn scaled by the factor n−1⁄2 converge in distribution to a standard normal distribution. By Kolmogorov's zero–one law, for any fixed M, the probability that the event occurs is 0 or 1. Then so An identical argument shows that This implies that these quantities cannot converge almost surely. In fact, they cannot even converge in probability, which follows from the equality and the fact that the random variables are independent and both converge in distribution to The law of the iterated logarithm provides the scaling factor where the two limits become different: Thus, although the absolute value of the quantity is less than any predefined ε > 0 with probability approaching one, it will nevertheless almost surely be greater than ε infinitely often; in fact, the quantity will be visiting the neighborhoods of any point in the interval (-1,1) almost surely. The law of the iterated logarithm (LIL) for a sum of independent and identically distributed (i.i.d.) random variables with zero mean and bounded increment dates back to Khinchin and Kolmogorov in the 1920s. Since then, there has been a tremendous amount of work on the LIL for various kinds of dependent structures and for stochastic processes.