On the Role of the Law of Large Numbers in the Theory of Randomness

In the first part of this article, we answer Kolmogorov's question (stated in 1963 in [1]) about exact conditions for the existence of random generators. Kolmogorov theory of complexity permits of a precise definition of the notion of randomness for an individual sequence. For infinite sequences, the property of randomness is a binary property, a sequence can be random or not. For finite sequences, we can solely speak about a continuous property, a measure of randomness. Is it possible to measure randomness of a sequence $t$ by the extent to which the law of large numbers is satisfied in all subsequences of $t$ obtained in an “admissible way”? The case of infinite sequences was studied in [2]. As a measure of randomness (or, more exactly, of nonrandomness) of a finite sequence, we consider the specific deficiency of randomness $\delta$ (Definition 5). In the second part of this paper, we prove that the function $\delta/\ln(1/\delta)$ characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied.