Purely Game-theoretic Random Sequences: I. Strong Law of Large Numbers and Law of the Iterated Logarithm

Abstract

Random sequences are usually defined with respect to a probability distribution~$\bf P$ (a $\sigma$-additive set function, normed to one, defined over a $\sigma$-algebra) assuming Kolmogorov's axioms for probability theory. In this paper, without using this axiomatics, we give a definition of random (typical) sequences taking as primitive the notion of a martingale and using the principle of the excluded gambling strategy. In this purely game-theoretic framework, no probability distribution or, partially or fully specified, system of conditional probability distributions needs to be introduced. For these typical sequences, we prove direct algorithmic versions of Kolmogorov's strong law of large numbers (SLLN) and of the upper half of Kolmogorov's law of the iterated logarithm~(LIL).