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We give a unified treatment of the convergence of random series and the rateof convergence of strong law of large numbers in the framework ofgame-theoretic probability of Shafer and Vovk (2001). We consider games withthe quadratic hedge as well as more general weaker hedges. The lattercorresponds to existence of an absolute moment of order smaller than two in themeasure-theoretic framework. We prove some precise relations between theconvergence of centered random series and the convergence of the series ofprices of the hedges. When interpreted in measure-theoretic framework, theseresults characterize convergence of a martingale in terms of convergence of theseries of conditional absolute moments. In order to prove these results wederive some fundamental results on deterministic strategies of Reality, who isa player in a protocol of game-theoretic probability. It is of particularinterest, since Reality's strategies do not have any counterparts inmeasure-theoretic framework, ant yet they can be used to prove results, whichcan be interpreted in measure-theoretic framework.

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