Levy's zero-one law in game-theoretic probability

Glenn Shafer, Vladimir Vovk, Akimichi Takemura

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We prove a non-stochastic version of Levy's zero-one law, and deduce several corollaries from it, including non-stochastic versions of Kolmogorov's zero-one law, the ergodicity of Bernoulli shifts, and a zero-one law for dependent trials. Our secondary goal is to explore the basic definitions of game-theoretic probability theory, with Levy's zero-one law serving a useful role.
Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalJournal of Theoretical Probability
Issue number1
Publication statusPublished - 2012


  • Doob’s martingale convergence theorem
  • ergodicity of Bernoulli shifts
  • Kolmogorov’s zero-one law
  • Levy’s martingale convergence theorem

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