The Law of the Iterated Logarithm in Analaysis

In this paper, we first discuss the history of the law of the iterated logarithm. We then focus our discussion on how it was introduced in analysis. Finally we mention different types of law of the iterated logarithm and state some of the recent developments. In order to discuss the history and developments of law of the iterated logarithm, some definitions and theorems are in order: Definition 1 [Normal Numbers] Suppose that N takes values in [0,1) and consider its decimal and dyadic expansion as: Now for a fixed k, , let denote the number of digits among the first n-digits of N that are equal to k. Then is the relative frequency of the digit k in the first n places and thus the limit is the frequency of the k in N. Then the number N is called the normal to the base 10 if and only if this limit exists for each k and is equal to . Similarly, the number N is called the normal to base 2 if and only if the limit exists and is equal to . Definition 2 [Lacunary Series] A real trigonometric series with partial sums which has is called q-lacunary series. In the above definition, the condition is called the gap condition (lacunarity condition) which states that the sequence { } increases at least as rapidly as a geometric progression whose common ratio is bigger than 1. Lacunary series exhibit many of the properties of partial sums of independent random variables. In the modern probability theor, lacunary series are called 'weakly dependent' random variables.