Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.

Behaviour of exact constants in inequalities of Jackson type

We studied the behaviour of exact constants in inequalities of Jackson type in $L_p$ space.

On inequalities of Kolmogorov type for classes of functions that are multiply monotonic on a finite interval

We solve the problem about exact constants in additive Kolmogorov-type inequalities on the class of multiply monotonic functions, defined on a finite interval.

On the Application of the Collocation Method in Spaces of p-Summable Functions to the Fredholm Integral Equation of the Second Kind

This work is devoted to the study of the numerical solution by the spline collocation method of the Fredholm equation of the second kind. For numerical solutions of such problems, the classical collocation method using polynomials is not always realizable in spaces of p-summable functions for numerical solutions of such problems. It is not always possible to obtain characteristics and estimates of errors of such approximations even in the case of its implementation. In this regard, in recent years, in practice, approximations are built using finite-difference methods. The purpose of this study is to obtain estimates of the error of the obtained approximate solution in the spaces indicated above. In addition, several statements were obtained about a pointwise estimate of this error at collocation nodes in terms of the kernel norm in specially constructed spaces of functions summable over the second variable. To obtain the main results, third-order collocation splines are used, as well as integral and averaged modules of smoothness. In this case, the results obtained can become a starting point for working with collocation splines of higher orders. In the case of the third order, the exact constants involved in the estimates are obtained. These results can be extended to the case of linear, parabolic collocation splines, as well as splines of order higher than the third.

Some asymptotic formulars for a Brownian motion whose dimension is variable with a regular variation from a parabolic domain to research the variation of biological species

Consider a Brownian motion with a regular variation starting at an interior point of a domain D in R d+1 ,d ≥ 1, let τ D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of logP(τ D > T) are given for T → ∞, depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the first exit time and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.

Upper and lower bounds of the first exit time of Brownian motion from The Minimum and Maximun Parabolic Domains with variable dimension to research the variation of biological species

Consider a Brownian motion with variable dimension starting at an interior point of the minimum or maximum parabolic domains Dmax t and Dmin t in Rd(t)+2, d(t) ≥ 1 is an increasing integral function as t →∞,d(t) →∞, and let τDmax t and τDmin t denote the first time the Brownian motion exits from Dmax t and Dmin t , respectively. Upper and lower bounds with exact constants for the asymptotics of logP(τDmax t > t) and logP(τDmin t > t) are given as t → ∞, depending on the shape of the domain Dmax t and Dmin t . The methods of proof are based on Gordon’s inequality and early works of Li, Lifshits and Shi in the single general parabolic domain case.

Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces Sp

In the paper, exact constants in direct and inverse approximation theorems for functions of several variables are found in the spaces Sp. The equivalence between moduli of smoothness and some K-functionals is also shown in the spaces Sp.