2005/10/31 by Masayuki Kumon, Akimichi Takemura, Kei Takeuchi · 1 citation
Mathematics · Economics, Econometrics and Finance · #math.ST #math.PR #q-fin.TR #stat.TH #msc:62M20 #msc:62C10
paper · pdf · doi:10.1080/07362990802405646
published as Stochastic Analysis and Applications,26:6,1161-1180,2008
arxiv created 2008/09/26 · arxiv updated 2009/12/01
We study capital process behavior in the fair-coin game and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O(√(log n/n)) and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.