Law of the Iterated Logarithm

Abstract

For sums of independent random variables we already know two limit theorems: the law of large numbers and the central limit theorem. The law of large numbers describes for large $n\in \mathbb{N}$ the typical behavior, or average value behavior, of sums of n random variables. On the other hand, the central limit theorem quantifies the typical fluctuations about this average value.

In Chapter 23, we will study atypically large deviations from the average value in greater detail. The aim of this chapter is to quantify the typical fluctuations of the whole process as n→∞. The main message is: While for fixed time n the partial sum S _n deviates by approximately $\sqrt{n}$ from its expected value (central limit theorem), the maximal fluctuation up to time n is of order $\sqrt{n \log \log n}$ (Hartman–Wintner theorem, Theorem 22.11).