Game-theoretic versions of strong law of large numbers for unbounded variables

摘要

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (Shafer, G. and Vovk, V. 2001, Probability and Finance: It's Only a Game! (New York: Wiley)). We prove several versions of SLLN for the case that Reality's moves are unbounded, Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results, However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges, In measure-theoretic proofs existence of moments is assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.