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On the Role of the Law of Large Numbers in the Theory of Randomnessстатья
Дата последнего поиска статьи во внешних источниках: 19 сентября 2015 г.
- Авторы: Muchnik Andrej, Semenov Alexei
- Журнал: Problems of Information Transmission
- Том: 39
- Номер: 1
- Год издания: 2003
- Издательство: Maik Nauka/Interperiodica Publishing
- Местоположение издательства: Russian Federation
- Первая страница: 119
- Последняя страница: 147
- DOI: 10.1023/A:1023638717091
- Аннотация: 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.
- Добавил в систему: Золин Евгений Евгеньевич