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. Another statement was given by A. N. Kolmogorov in 1929.
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−½converge 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 quantity is less than any predefined ε > 0 with probability approaching one, the quantity will nevertheless be greater thanε infinitely often; in fact, the quantity will be visiting the neighborhoods of any point in the interval almost surely.
Generalizations and variants
The law of the iterated logarithm for a sum of independent and identically distributed 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. Following is a small sample of notable developments. Hartman–Wintner generalized LIL to random walks with increments with zero mean and finite variance. Strassen studied LIL from the point of view of invariance principles. Stout generalized the LIL to stationary ergodic martingales. De Acosta gave a simple proof of Hartman–Wintner version of LIL. Wittmann generalized Hartman–Wintner version of LIL to random walks satisfying milder conditions. Vovk derived a version of LIL valid for a single chaotic sequence. This is notable, as it is outside the realm of classical probability theory. Yongge Wang has shown that the law of the iterated logarithm holds for polynomial time pseudorandom sequences also. The Java-based software tests whether a pseudorandom generator outputs sequences that satisfy the LIL. A non-asymptotic version that holds over finite-time martingale sample paths has also been proved and applied.
