No. 984

Title:

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

Author(s):

Masayuki, Kumon (Risk Analysis Research Center, Institute of Statistical Mathematics);
Akimichi, Takemura (Graduate School of Information Science and Technology, University of Tokyo);
Kei, Takeuchi (Faculty of International Studies, Meiji Gakuin University)

Key words:

Borel-Cantelli lemma, call option, Doob's upcrossing lemma, Kronecker's lemma, Marcinkiewicz-Zygmund strong law, martingale convergence theorem.

Abstract:

    We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (2001). 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 are assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.