On-Line Probability, Complexity and Randomness

  • Alexey Chernov
  • Alexander Shen
  • Nikolai Vereshchagin
  • Vladimir Vovk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5254)


Classical probability theory considers probability distributions that assign probabilities to all events (at least in the finite case). However, there are natural situations where only part of the process is controlled by some probability distribution while for the other part we know only the set of possibilities without any probabilities assigned.

We adapt the notions of algorithmic information theory (complexity, algorithmic randomness, martingales, a priori probability) to this framework and show that many classical results are still valid.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gács, P.: Lecture notes on descriptional complexity and randomness,
  2. 2.
    Levin, L.A.: Uniform tests of randomness. Soviet Math. Dokl. 17(2), 337–340 (1976)zbMATHGoogle Scholar
  3. 3.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 2nd edn. Springer, New York (1997)zbMATHGoogle Scholar
  4. 4.
    Muchnik An, A., Chernov, A., Shen, A.: Algorithmic randomness and splitting of supermartingales, Scholar
  5. 5.
    Shen, A.: Algorithmic information theory and Kolmogorov complexity, Technical Report 2000-034. Uppsala Universitet publication,
  6. 6.
    Uspensky, V.A., Semenov, A.L., Shen, A.: Can an individual sequence of zeros and ones be random? Russian Mathematics Surveys 45, 121–189 (1990)CrossRefGoogle Scholar
  7. 7.
    Shafer, G., Vovk, V.: Probability and Finance: It’s Only a Game. Wiley, New York (2001)zbMATHGoogle Scholar
  8. 8.
    Vovk, V., Shen, A.: Prequential randomness. In: Freund, Y., Györfi, L., Turán, G., Zeugmann, T. (eds.) ALT 2008. LNCS(LNAI), vol. 5254, pp. 154–168. Springer, Heidelberg (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Alexey Chernov
    • 1
  • Alexander Shen
    • 2
  • Nikolai Vereshchagin
    • 3
  • Vladimir Vovk
    • 1
  1. 1.Royal HollowayUniversity of LondonEghamUK
  2. 2.Marseille and Institute of Information Transmission ProblemsLIF (Université Aix-Marseille & CNRS)Moscow
  3. 3.Moscow State University 

Personalised recommendations