Theory of approximation and continuity of random processes

TL;DR

This paper introduces degenerate approximation for multivariable functions and applies it to analyze the local structure of random processes, providing conditions for their continuity, compactness, and convergence.

Contribution

It develops a new notion of approximation for multivariable kernels and applies it to establish criteria for the continuity and convergence of random processes.

Findings

Necessary and sufficient conditions for Gaussian process continuity

Conditions for weak compactness of random process families

Results on convergence and the Central Limit Theorem in continuous function spaces

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.