On some aspects of the prequential and algorithmic approaches to probability and statistical theory.

Following an axiomatic introduction to the prequential (predictive sequential) principle to statistical inference proposed by A. P. Dawid, in which we consider some of the questions it raises, we examine a conjecture on the supposed prequential asymptotic behaviour of significance levels based on a particular class of test statistics. Then, after a presentation of some martingale probability frameworks recently proposed by V. G. Vovk, algorithmic constraints are introduced to give a definition of random sequences on the lines of Martin-Lof's classical approach. This definition, instead of being given, as in the classical algorithmic approach, with respect to a Kolmogorovian probability distribution P, is given only with respect to a sequence of measurable functions by using the principle of the excluded gambling strategy. The idea underlying this approach is that if we are to play an infinite sequence of fair games against an infinitely rich bookmaker, then, whatever computable strategy we choose, we shall never become richer and richer as the game goes on. These random sequences, apart from some basic properties, have been shown to satisfy: an analogue of Kolmogorov's strong law of large numbers; an analogue of the upper half of Kolmogorov's law of the iterated logarithm for binary martingales; and an analogue of Schatte's strong central limit theorem for the coin-tossing process. Besides, for these random sequences, we also investigated the distribution of the values of the corresponding infinite single realizations, in the case of two basic processes. These last results, together with the strong central limit theorem, would provide an instance in which 'empirical' distribution functions are derived without the assumption of any Kolmogorovian probability distribution