Journal of Theoretical Probability

, Volume 25, Issue 1, pp 1–24

Lévy’s Zero–One Law in Game-Theoretic Probability



We prove a nonstochastic version of Lévy’s zero–one law and deduce several corollaries from it, including nonstochastic versions of Kolmogorov’s zero–one law and the ergodicity of Bernoulli shifts. Our secondary goal is to explore the basic definitions of game-theoretic probability theory, with Lévy’s zero–one law serving a useful role.


Doob’s martingale convergence theorem Ergodicity of Bernoulli shifts Kolmogorov’s zero–one law Lévy’s martingale convergence theorem 

Mathematics Subject Classification (2000)

60F20 60G42 60A05 

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Glenn Shafer
    • 1
    • 2
  • Vladimir Vovk
    • 2
  • Akimichi Takemura
    • 3
  1. 1.Department of Accounting and Information SystemsRutgers Business School—Newark and New BrunswickNewarkUSA
  2. 2.Department of Computer ScienceRoyal Holloway, University of LondonEghamEngland
  3. 3.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoBunkyo-ku, TokyoJapan

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