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.

Criterion for convergence almost everywhere, with applications

“…Immediate predecessor of general case besides the special cases: martingales, monotone sequences etc. is the preprint [36]. We intend to generalize the results obtained therein.…”

“…The particular case of this problem is the problem of continuity of random processes and fields, is considered in the articles and books [10], [11], [12], [13], [16], [17], [21], [24], [26], [27], [28], [29], [32], [33], [36], [41], [42], [43], [44], [45], [46], [48] etc.…”

CLT for continuous random processes under approximations terms

“…This problem contains as a subproblem the finding of a sufficient condition for continuity with probability one of random processes, Gaussian or not, see [2] - [5], [12] - [14], [16], [20], [21], [24], [22]- [27], [31] - [35], [38] - [41], [43], [47], [49] - [53], [55].…”

“…Note that in the articles [11], [23], [22], [30] the CLT is formulated in the so-called metric entropy terms, in [11], [20], [37], [43] by means of the notions majorizing measure.…”

“…Remark 2.1. The conditions of theorem 2.1 are essentially non-improvable still for the Gaussian processes, see [12], [27], [40], [41], [42], [43].…”