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Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability

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  • Sato, Ryosuke
  • Miyabe, Kenshi
  • Takemura, Akimichi

Abstract

We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We establish the relation between the rate of convergence of the strong law of large numbers in the self-normalized form and the rate of divergence to infinity of the prior density around the origin. In particular we present prior densities ensuring the validity of Erdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm.

Suggested Citation

  • Sato, Ryosuke & Miyabe, Kenshi & Takemura, Akimichi, 2018. "Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1466-1484.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:5:p:1466-1484
    DOI: 10.1016/j.spa.2017.07.014
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    References listed on IDEAS

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    4. Miyabe, Kenshi & Takemura, Akimichi, 2015. "Derandomization in game-theoretic probability," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 39-59.
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    6. Miyabe, Kenshi & Takemura, Akimichi, 2013. "The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3132-3152.
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