No abstract

In this paper non-asymptotic moment estimates are derived for tail of distribution for discrete time polynomial martingale by means of martingale differences as a rule in the terms of unconditional and unconditional relative moments and tails of distributions of summands.We show also the exactness of obtained estimations.

We offer in this paper the non-asymptotical bilateral sharp exponential estimates for tail of maximum distribution of discontinuous random fields.Our consideration based on the theory of Prokhorov-Skorokhod spaces of random fields and on the theory of multivariate Banach spaces of random variables with exponential decreasing tails of distributions.

Motivated by finance and technical applications, the objective of this paper is to consider adaptive estimation of regression and density distribution based on Fourier-Legendre expansion, and construction of confidence intervals -also adaptive. The estimators are asymptotically optimal and adaptive in the sense that they can adapt to unknown smoothness.

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.