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On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

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Abstract

In the framework of the game-theoretic probability of Shafer and Vovk it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers in the bounded forecasting game. We present a simple finite-memory strategy based on the past average of Reality’s moves, which weakly forces the strong law of large numbers with the convergence rate of \({O(\sqrt{\log n/n})}\) . Our proof is very simple compared to a corresponding measure-theoretic result of Azuma (Tôhoku Mathematical Journal, 19, 357–367, 1967) on bounded martingale differences and this illustrates effectiveness of game-theoretic approach. We also discuss one-sided protocols and extension of results to linear protocols in general dimension.

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Authors and Affiliations

  1. Risk Analysis Research Center, Institute of Statistical Mathematics, 4-6-7 Minami-azabu, Minato-ku, Tokyo, 106-8569, Japan

    Masayuki Kumon

  2. Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Akimichi Takemura

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  1. Masayuki Kumon
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  2. Akimichi Takemura
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Correspondence to Akimichi Takemura.

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Kumon, M., Takemura, A. On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game. Ann Inst Stat Math 60, 801–812 (2008). https://doi.org/10.1007/s10463-007-0125-5

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  • DOI: https://doi.org/10.1007/s10463-007-0125-5

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