Random Sequences

Abrahamic Reflections on Randomness and Providence, 2022

While ‘random’ is a familiar word, it takes on a variety of definitions across mathematics and the sciences. This paper gives an overview of the terrain, looking at how randomness and closely related terms are used in a variety of disciplines. As will see, some are more relevant to the question of providence than others.

Theoretical Computer Science, 2003

In this paper, we investigate reÿned deÿnition of random sequences. Classical deÿnitions (Martin-L of tests of randomness, uncompressibility, unpredictability, or stochasticness) make use of the notion of algorithm. We present alternative deÿnitions based on set theory and explain why they depend on the model of ZFC that is considered. We also present a possible generalization of the deÿnition when small inÿnite regularities are allowed. (B. Durand), kanovei@mccme.ru (V. Kanovei), uspen-sky@lpcs.math.msu.ru (V.A. Uspensky), ver@mech.math.msu.su (N. Vereshchagin).

2003

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.

Resonance, 1996

Mathematical Metaphysics of Randomness // Theoretical Computer Science. 207:2, 1998

We consider various mathematical refinements of the notion of randomness of an infinite sequence and relations between them. On the base of the game approach a new possible refinement of the notion of randomness is introduced - unpredictability. The case of finite sequences is also considered.

Lecture Notes in Computer Science, 2012

This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed.

Russian Mathematical Surveys. 45:1, 1990

How can one define that an individual binary sequence is random? The paper presents a comparative study of different approaches to this problem. The three properties of randomness to be analysed are: stochastic, chaotic, and typic (i.e., random in the senses given by von Mises, Kolmogorov, and Martin-Löf, respectively). The randomness of a (binary) sequence is to be considered with respect to some probability distribution over the set of all the sequences, e.g., the uniform Bernoulli distribution. The concept underlying the typicness is that of effectively zero sets of sequences: a sequence is said to be typic if it is not contained in any effectively zero set of sequences. The concept underlying the chaoticness is that of monotone entropy (complexity) KM(x) of any given finite sequence x: a sequence is said to be chaotic if KM(x)=−logP(Ωx)+O(1) for any initial segment x of this sequence. (Ωx is the set of all infinite sequences initiated by x, and P(Ωx) is the measure of this set in the given probability distribution.) The Levin-Schnorr theorem shows that a sequence is typic (w.r.t. the uniform Bernoulli distribution) iff it is chaotic. A sequence x is stochastic (i.e., random in the von Mises’ sense) w.r.t the uniform Bernoulli distribution if for any infinite subsequence of x chosen by some “admissible rule of choice”, the frequency of occurrence of the symbol 1 in initial segments of this subsequence tends to 1/2 as the lengths of these segments increase infinitely. Here the notion “admissible rule of choice” is requested to be defined. Two definitions of this notion introduced by Church and by Kolmogorov-Loveland are considered, and, according to them two definitions of the notion “stochastic sequence” are obtained. Unfortunately, these two definitions are not equivalent, and every of them is weaker than that of a typic, or chaotic, sequence.

Problemy Peredachi Informatsii, 39:1, 2003

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.

Problems of Information Transmission. 39:1, 2003

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.