Journal of Theoretical Probability

, Volume 25, Issue 1, pp 1–24 | Cite as

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

  • Glenn Shafer
  • Vladimir VovkEmail author
  • Akimichi Takemura


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 



Our thinking about Lévy’s zero–one law was influenced by a preliminary draft of [2]. We are grateful to Gert de Cooman for his questions that inspired some of the results in Sect. 3 of this article. This article has benefitted very much from a close reading by its anonymous referee, whose penetrating comments have led to a greatly improved presentation (in particular, Lemmas 1, 2, 4, 5 and the final statements of Theorem 2 and Lemma 8 are due to him or her) and helped us correct a vacuous statement in a previous version. Andrzej Ruszczyński has brought to our attention the literature on coherent measures of risk. Our work has been supported in part by EPSRC grant EP/F002998/1.


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Glenn Shafer
    • 1
    • 2
  • Vladimir Vovk
    • 2
    Email author
  • 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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