An exponential inequality and the convergence rate of the strong law of large numbers in the unbounded forecasting game

Abstract

In this paper, we study a large deviation inequality in the framework of the game-theoretic probability of Shafer and Vovk (Probability and Finance: It's only a game! Wiley, 2001). First, we propose an exponential inequality in the unbounded forecasting game. In addition, by using the exponential inequality, we prove that in the unbounded forecasting game, under some conditions, Skeptic can force the strong law of large numbers with the convergence rate of .

Keywords:

• exponential inequality
• game-theoretic probability
• strong law of large numbers
• convergence rate

AMS Subject Classification::

• 60F10
• 60F15
• 91A60

Acknowledgements

The author thanks an anonymous referee for very careful reading of the manuscript and for many valuable comments which improved the paper. In particular, the author wishes to thank the referee for insightful suggestions that strengthened Theorem 3.1 and simplified its proof.