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 strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk [24]. 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.

Bayesian Logistic Betting Strategy Against Probability Forecasting

“…However we could not succeed to prove weak forcing of θ * n → 0 by SOS alone. For the case of d = 1 we could employ approaches of [14] to prove results similar to Theorem 4.2. In [14] we also discussed Reality's strategies.…”

Erdős–Feller–Kolmogorov–Petrowsky law of the iterated logarithm for self-normalized martingales: A game-theoretic approach

“…This is easily seen by dividing the infinite series into blocks of sums less than or equal to 1/2 k and multiplying the k-th block by k (see also [13,Lemma 4.15]). For k ≥ 1 let…”

The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges

“…Definition 2.12 (Miyabe and Takemura [6]). By a strategy R, Reality complies with the event E if (i) irrespective of the moves of Forecaster and Skeptic, both observing their collateral duties, E happens, and (ii) sup n K n < ∞.…”

“…Theorem 2.13 (Miyabe and Takemura [6]). In the unbounded forecasting, if Skeptic can force an event E, then Reality complies with E.…”