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

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Shafer, Glenn; Vovk, Vladimir; Takemura, Akimichi;
(2009)
  • Subject: Mathematics - Probability | 60F20

We prove a game-theoretic 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 explor... View more
  • References (11)
    11 references, page 1 of 2

    [1] Pal Bartfai and Pal Revesz. On a zero-one law. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 7:43{47, 1967.

    [2] Bernard Bru and Salah Eid. Jessen's theorem and Levy's lemma: A correspondence. Electronic Journal for History of Probability and Statistics, 5(1), June 2009. Available on-line at http://www.jehps.net/.

    [3] Isaac P. Cornfeld, Sergei V. Fomin, and Yakov G. Sinai. Ergodic Theory. Springer, New York, 1982.

    [4] Gert de Cooman and Filip Hermans. Imprecise probability trees: bridging two theories of imprecise probability. Arti cial Intelligence, 172:1400{1427, 2008.

    [5] J rgen Ho mann-J rgensen. The general marginal problem. In Svetozar Kurepa, Hrvoje Kraljevic, and Davor Butkovic, editors, Functional Analysis II, volume 1242 of Lecture Notes in Mathematics, pages 77{367. Springer, Berlin, 1987.

    [6] Andrei N. Kolmogorov. Grundbegri e der Wahrscheinlichkeitsrechnung. Springer, Berlin, 1933. English translation: Foundations of the Theory of Probability. Chelsea, New York, 1950.

    [7] Paul Levy. Theorie de l'addition des variables aleatoires. Gauthier-Villars, Paris, 1937. Second edition: 1954.

    [8] Glenn Shafer and Vladimir Vovk. Probability and Finance: It's Only a Game! Wiley, New York, 2001.

    [9] Akimichi Takemura, Vladimir Vovk, and Glenn Shafer. The generality of the zero-one laws. Accepted for publication and published on-line in Annals of the Institute of Statistical Mathematics. Available as arXiv technical report arXiv:0803.3679 [math.PR], 2009.

    [10] Vladimir Vovk. Prequential probability: game-theoretic = measuretheoretic. The Game-Theoretic Probability and Finance project, Working Paper 27, http://probabilityandfinance.com, http://arxiv. org/abs/0905.1673, May 2009. Conference version: JSM 2009. The journal version is to appear in Theoretical Computer Science.

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