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

TL;DR

This paper unifies the analysis of random series convergence and the strong law of large numbers' rate of convergence within game-theoretic probability, linking hedge prices and martingale convergence.

Contribution

It introduces a unified approach to convergence analysis in game-theoretic probability, including new results on deterministic strategies of Reality and their measure-theoretic implications.

Findings

Relations between convergence of random series and hedge price series

Characterization of martingale convergence via absolute moments

Fundamental results on Reality's deterministic strategies

Abstract

We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001). We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two 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 measure-theoretic framework, these results characterize convergence of a martingale in terms of 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 measure-theoretic framework, ant yet they can be used to prove results, which can be interpreted in measure-theoretic framework.