Theory of approximation and continuity of random processes

Abstract

We introduce and investigate a new notion of the theory of approximation-the so-called degenerate approximation, i.e. approximation of the function of two (and more) variables (kernel) by means of degenerate function (kernel). We apply obtained results to the investigation of the local structure of random processes, for example, we find the necessary and sufficient condition for continuity of Gaussian and non-Gaussian processes, some conditions for weak compactness and convergence of a family of random processes, in particular, for Central Limit Theorem in the space of continuous functions. We give also many examples in order to illustrate the exactness of proved theorems.