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

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 δ (Definition 5). In the second part of this paper, we prove that the function δ/ln(1/δ) characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

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

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Kluwer Academic Publishers-Plenum Publishers
Copyright © 2003 by MAIK “Nauka/Interperiodica”
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
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