Kolmogorov's Contributions to the Foundations of Probability

Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical and philosophical foundations of probability in the twentieth century, and his thinking on the topic is still potent today. In this article we first review the three stages of Kolmogorov's work on the foundations of probability: (1) his formulation of measure-theoretic probability, 1933; (2) his frequentist theory of probability, 1963; and (3) his algorithmic theory of randomness, 1965–1987. We also discuss another approach to the foundations of probability, based on martingales, which Kolmogorov did not consider.

[1]  V. Vovk.  The Law of the Iterated Logarithm for Random Kolmogorov, or Chaotic, Sequences , 1988 .

[2]  A. Church.  On the concept of a random sequence , 1940 .

[3]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1997, Texts in Computer Science.

[4]  C. Schnorr.  Zufälligkeit und Wahrscheinlichkeit , 1971 .

[5]  G. Shafer,et al.  Probability and Finance: It's Only a Game! , 2001 .

[6]  Vladimir V. V'yugin,et al.  Nonrobustness Property of the Individual Ergodic Theorem , 2001, Probl. Inf. Transm..

[7]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part I , 1964, Inf. Control..

[8]  A. N. Kolmogorov.  Combinatorial foundations of information theory and the calculus of probabilities , 1983 .

[9]  A. P. Dawid,et al.  Present position and potential developments: some personal views , 1984 .

[10]  J. Lukasiewicz.  Logical foundations of probability theory , 1970 .

[11]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

[12]  Claus-Peter Schnorr,et al.  A unified approach to the definition of random sequences , 1971, Mathematical systems theory.

[13]  J. Runnenburg.  PROBABILITY THEORY AND ITS APPLICATIONS , 1985 .

[14]  E. Asarin.  Some Properties of Kolmogorov $\Delta$-Random Finite Sequences , 1988 .

[15]  Vladimir Vovk.  Kolmogorov’s Complexity Conception of Probability , 2001 .

[16]  Michel Loève,et al.  Probability Theory I , 1977 .

[17]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[18]  R. Solomonoff.  A PRELIMINARY REPORT ON A GENERAL THEORY OF INDUCTIVE INFERENCE , 2001 .

[20]  C. Schnorr.  Klassifikation der Zufallsgesetze nach Komplexität und Ordnung , 1970 .

[21]  A. Kolmogorov.  On Logical Foundations of Probability Theory , 1983 .

[22]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[23]  Yu. A. Gur'yan,et al.  Parts I and II , 1982 .

[24]  Jean-Luc Ville.  Étude critique de la notion de collectif , 1939 .

[25]  J. Doob.  Stochastic processes , 1953 .

[26]  Claus-Peter Schnorr,et al.  Process complexity and effective random tests , 1973 .

[27]  Péter Gács,et al.  Algorithmic statistics , 2000, IEEE Trans. Inf. Theory.

[28]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[29]  Andrei N. Kolmogorov,et al.  On Tables of Random Numbers (Reprinted from "Sankhya: The Indian Journal of Statistics", Series A, Vol. 25 Part 4, 1963) , 1998, Theor. Comput. Sci..

[30]  A. Dawid.  Calibration-Based Empirical Probability , 1985 .

[31]  V. V'yugin.  On the Defect of Randomness of a Finite Object with Respect to Measures with Given Complexity Bounds , 1988 .

[32]  Richard Von Mises,et al.  Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik , 1931 .

[33]  Nikolai K. Vereshchagin,et al.  Kolmogorov's structure functions with an application to the foundations of model selection , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[34]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .