Kolmogorov's Law of Large Numbers
Glenn R. Shafer, Vladimir Vovk · Wiley series in probability and statistics · 2019
This chapter presents the game-theoretic version of Kolmogorov's law more precisely. Like the proof of the law of large numbers for bounded outcomes, the proof of Kolmogorov's law for unbounded outcomes will use a simplified protocol. The chapter provides the construction of Skeptic's strategy, which requires a game-theoretic version of Doob's convergence theorem. It explores the proof of the game-theoretic version of Kolmogorov's law by constructing a strategy for Skeptic and a strategy for Reality. The chapter considers different ways of describing the game between Skeptic and Reality when one tries to make an event happen and the other tries to make it fail. Skeptic can force an event if and only if there is a nonnegative supermartingale that tends to infinity on paths where the event fails. The chapter utilizes Martin's theorem on the determinacy of perfect-information games to gain more insight into the relationship between forcing by Skeptic and forcing by Reality.