No. 951

Title:

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

Author(s):

Masayuki, Kumon (Risk Analysis Reserch Center, Institute of Statistical Mathematics);
Akimichi, Takemura (Graduate School of Information Science and Technology, University of Tokyo)

Key words:

Azuma-Hoeffding-Bennett inequality; capital process; game-theoretic probability; Kullback divergence; large deviation.

Abstract:

    In the framework of the game-theoretic probability of Shafer and Vovk (2001) 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 $O(\sqrt{\log n/n})$. We also give a detailed analysis of the paths of Skeptic's capital process for the case of the fair-coin game when our strategy is used. We show that if Reality violates SLLN, then the exponential growth rate of Skeptic's capital process is explicitly 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.