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Takemura, Akimichi; Vovk, Vladimir; Shafer, Glenn

The generality of the zero-one laws. (English) Zbl 1234.60037

Ann. Inst. Stat. Math. 63, No. 5, 873-885 (2011).
This paper deals with the generalization of three zero-one laws in the game-theoretic framework, as introduced by G. Shafer and V. Vovk [Probability and finance: it’s only a game! Chichester: Wiley (2001; Zbl 0985.91024)]. More precisely, the authors use some simple martingale arguments in order to generalize Kolmogorov’s zero-one law (see, e.g., [A. N. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebnisse der Math. 2, Nr. 3 (1933; JFM 59.1152.03); Foundations of the theory of probability. - Translated by N. Morrison. New York: Chelsea Publishing Company (1950; Zbl 0037.08101)]), ergodicity of Bernoulli shifts [I. P. Cornfeld et al., Ergodic theory. Transl. from the Russian by A. B. Sossinskii. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 0493.28007)] and Hewitt and Savage’s zero-one law [E. Hewitt and L. J. Savage, Trans. Am. Math. Soc. 80, 470–501 (1955; Zbl 0066.29604)]. To this end, after laying out the protocols corresponding to each game, they define upper and lower probabilities in a zero-one law context, and they present their martingale proofs.
Reviewer: Stavros Vakeroudis (Paris)

Cited in 6 Documents

MSC:

60F20 Zero-one laws
60G44 Martingales with continuous parameter
91A60 Probabilistic games; gambling
91A05 2-person games

Keywords:

zero-one laws; game-theoretic probability; invariant event; martingale; permutable event; tail event; upper probability; lower probability

Citations:

Zbl 0985.91024; Zbl 0037.08101; Zbl 0493.28007; Zbl 0066.29604; JFM 59.1152.03
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\textit{A. Takemura} et al., Ann. Inst. Stat. Math. 63, No. 5, 873--885 (2011; Zbl 1234.60037)
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References:

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