×

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

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.

MSC:

60F20 Zero-one laws
60G44 Martingales with continuous parameter
91A60 Probabilistic games; gambling
91A05 2-person games
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bártfai P., Révész P. (1967) On a zero-one law. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 7: 43–47 · Zbl 0147.16904
[2] Chow Y.S., Robbins H., Siegmund D. (1971) Great expectations: The theory of optimal stopping. Houghton Mifflin, Boston · Zbl 0233.60044
[3] Cornfeld I.P., Fomin S.V., Sinai Y.G. (1982) Ergodic theory. Springer, New York · Zbl 0493.28007
[4] Hewitt E., Savage L.J. (1955) Symmetric measures on Cartesian products. Transactions of the American Mathematical Society 80: 470–501 · Zbl 0066.29604
[5] Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer. English translation (1950): Foundations of the theory of probability. New York: Chelsea.
[6] Shafer G., Vovk V. (2001) Probability and finance: It’s only a game!. Wiley, New York · Zbl 0985.91024
[7] Shiryaev A.N. (1996) Probability (2nd ed). Springer, New York · Zbl 0909.01009
[8] Takeuchi K. (2004) Mathematics of betting and financial engineering (in Japanese). Saiensusha, Tokyo
[9] Williams D. (1991) Probability with martingales. Cambridge University Press, Cambridge · Zbl 0722.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.