Search

In Part I of this paper [M. Minozzo, Theory Probab. Appl., 44 (1999), pp. 511-5221 a definition of typical sequences was given, without using any Kolmogorovian probability distribution P, by applying the principle of the excluded gambling strategy directly to a sequence of measurable functions. In this paper we forward this theory by deriving for these typical sequences some elementary limiting empirical distribution functions and some strong central limit theorem type results for the coin tossing process. [ABSTRACT FROM AUTHOR]

Random sequences are usually defined with respect to a probability distribution P (σ -additive set function, normed to one, defined over a σ-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). [ABSTRACT FROM AUTHOR]