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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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    1. Miyabe, Kenshi & Takemura, Akimichi, 2015. "Derandomization in game-theoretic probability," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 39-59.
    2. Erik Ordentlich & Thomas M. Cover, 1998. "The Cost of Achieving the Best Portfolio in Hindsight," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 960-982, November.
    3. Masayuki Kumon & Akimichi Takemura, 2008. "On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 801-812, December.
    4. Wang, Qiying, 1999. "Kolmogrov and Erdös test for self-normalized sums," Statistics & Probability Letters, Elsevier, vol. 42(3), pages 323-326, April.
    5. 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.
    6. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29, January.
    7. Kumon, Masayuki & Takemura, Akimichi & Takeuchi, Kei, 2011. "Sequential optimizing strategy in multi-dimensional bounded forecasting games," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 155-183, January.
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