Purely game-theoretic random sequences: II. Limiting empirical distributions and strong central limit theorem

In Part I of this paper [M. Minozzo, Theory Probab. Appl., 44 (1999), pp. 511–522] 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.