Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Abstract

We give a unified treatment of the convergence of random series and the rate of convergence of the strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001) [24]. We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to the existence of an absolute moment of order smaller than 2 in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in the measure-theoretic framework, these results characterize the convergence of a martingale in terms of the convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality’s strategies do not have any counterparts in the measure-theoretic framework, ant yet they can be used to prove results which can be interpreted in the measure-theoretic framework.