arXiv subjects

Oleksandr Mokliachuk

Publications and source records attributed to Oleksandr Mokliachuk.

5 recordsLinked to original sources

arXiv2026

Quasi-Banach spaces of random variables and stochastic processes

This book develops the theory of quasi-Banach $K_σ$-spaces $\mathbf{F}_ψ(Ω)$, $\mathbf{F}_ψ^*(Ω)$, and $D_{V,W}(Ω)$ of random variables and stochastic processes, extending the classical framework of Orlicz spaces, $Sub_φ(Ω)$ and $V(φ,ψ)$ spaces. The book consists of eleven chapters. The first two chapters establish the foundational theory: stochastic processes from quasi-Banach $K_σ$-spaces are introduced, and the fundamental properties of $\mathbf{F}_ψ(Ω)$ are studied in detail. The third chapter derives distribution estimates for suprema of processes from $\mathbf{F}_ψ^*(Ω)$, and the fourth addresses approximation theory in $SF_ψ(Ω)$. The fifth chapter examines Orlicz spaces and their connections to $\mathbf{F}_ψ(Ω)$. Chapters six and seven treat the pre-Banach $K_σ$-spaces $D_{V,W}(Ω)$, establishing their essential properties and evaluating reliability and accuracy of stochastic process models. The eighth chapter provides norm distribution estimates in $L_p(T)$ for processes from $\mathbf{F}_ψ(Ω)$. The ninth chapter develops the Monte Carlo method for multiple integrals over $\mathbb{R}^n$ with prescribed reliability and accuracy. The final two chapters treat modeling of $Sub_φ(Ω)$ processes - subclasses of $K_σ$-spaces - with given reliability and accuracy in $L_p(T)$ and $C(T)$ respectively. The results are substantially based on the authors' original work and that of their co-authors.

math.ST↗

arXiv2026

Estimation of reliability and accuracy of models of $φ$-sub-Gaussian process using generating functions of polynomial expansions

Stochastic processes are often represented through orthonormal series expansions, a framework originating in the classical works of Loève and Karhunen and widely used for simulation and numerical approximation. While truncation error in such expansions has been extensively studied, practical models frequently involve an additional source of error arising from the approximation of coefficient functions when closed-form expressions are unavailable. The combined effect of these two errors remains insufficiently addressed in the literature. Building on the author's earlier work on reliability and accuracy estimates for $φ$-sub-Gaussian processes, this paper extends the methodology to orthonormal polynomial systems that do not possess normalized generating functions in analytical form, including the Legendre, generalized Laguerre, and Gegenbauer families. New bounds are derived for models in $L_p(T)$ space that simultaneously account for truncation and coefficient approximation. The resulting criteria provide practical guidance for selecting the number of series terms required to achieve prescribed levels of reliability and accuracy across a broader class of polynomial-based stochastic process models.

math.ST↗

arXiv2025

Modelling with given reliability and accuracy in the space $L_p(T)$ of stochastic processes from $Sub_φ(Ω)$ decomposable in series with independent elements

Models that approximate stochastic processes from $Sub_φ(Ω)$ with given reliability and accuracy in $L_p(T)$ for some given $φ(t)$ are considered. We also study construction of models of processes which can be decomposed into series with approximate elements. Karhunen-Lo{è}ve model is considered as an example of the application of the proposed construction.

math.PR↗

arXiv2025

Estimation of accuracy and reliability of models of $φ$-sub-Gaussian stochastic processes in $C(T)$ spaces

At present, in the theory of stochastic process modeling a problem of assessment of reliability and accuracy of stochastic process model in $C(T)$ space wasn't studied for the case of implicit decomposition of process in the form of a series with independent terms. The goal is to study reliability and accuracy in $C(T)$ of models of processes from $Sub_φ(Ω)$ that cannot be decomposed in a series with independent elements explicitly. Using previous research in the field of modeling of stochastic processes, assumption is considered about possibility of decomposition of a stochastic process in the series with independent elements that can be found using approximations. Impact of approximation error of process decomposition in series with independent elements on reliability and accuracy of modeling of stochastic process in $C(T)$ is studied. Theorems are proved that allow estimation of reliability and accuracy of a model in $C(T)$ of a stochastic process from $Sub_φ(Ω)$ in the case when decomposition of this process in a series with independent elements can be found only with some error, for example, using numerical approximations.

math.ST↗

arXiv2025

Modeling of stochastic processes in $L_p(T)$ using orthogonal polynomials

In this paper, models that approximate stochastic processes from the space $Sub_φ(Ω)$ with given reliability and accuracy in $L_p(T)$ are considered for some specific functions $φ(t)$. For processes that are decomposited in series using orthonormal bases, such models are constructed in the case where elements of such decomposition cannot be found explicitly.

math.ST↗