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.

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• Exact exponential bounds for the random field maximum distribution via the majoring measures (generic chaining)
• Asymptotic exponential bounds for MLE deviation under minimal conditions via classical and generic chaining methods
• Module of continuity for the functions belonging to the Sobolev-Grand Lebesgue Spaces
• Continuity of functions belonging to the fractional order Sobolev-Grand Lebesgue Spaces
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Cited by in corpus (1)

• Moment and tail estimates and Banach space valued Non-Central Limit Theorem (NCLT) for sums of multi-indexed random variables, processes and fields