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

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

2005-08-11

arxiv.org/abs/math/0508190v3

Annals of the Institute of Statistical Mathematics, Vol.60, No.4, 801-812. (2008)

Mathematics

Probability

14 pages

Scientific paper

10.1007/s10463-007-0125-5

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 (SLLN) 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 (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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