Olga Vasylyk

National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” · Department of Mathematical Analysis and Probability Theory

Even though the problem of modelling and simulation is not new it continues to be actual over time. Our computers are becoming more powerful and this allows us to use more sofisticated algorithms for more complicated problems. In this paper we constructed the model from the series decomposition of the Gaussian stationary Ornstein–Uhlenbeck process....

In this paper, we consider random variables and stochastic processes from the space Fψ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{F}}_\psi (\Omega )$$\e...

The paper presents bounds for the distributions of suprema for particular classes of ϕ-sub-Gaussian random fields. Results stated depend on representations of bounds for increments of the fields in different metrics. Several examples of applications are provided to illustrate the results, in particular, to random fields related to stochastic partia...

In the paper, we consider random variables and stochastic processes from the space Fψ(Ω) and study approximation problems for such processes. The method of series decomposition of stochastic processes from Fψ(Ω) is used to find an approximating process called a model. The rate ofconvergence of the model to the process in the uniform norm is investi...

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such...

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are wi...

In this paper, there are studied sample paths properties of stochastic processes representing solutions of higher-order dispersive equations with random initial conditions given by φ-sub-Gaussian harmonizable processes. The main results are the bounds for the rate of growth of such stochastic processes considered over unbounded domains. The class o...

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are wid...

In this paper, we continue to study the properties of a separable strictly φ-sub-Gaussian quasi shot noise process $X(t) = \int_{-\infty}^{+\infty} g(t,u) d\xi(u), t\in\R$, generated by the response function g and the strictly φ-sub-Gaussian process ξ = (ξ(t), t ∈ R) with uncorrelated increments, such that E(ξ(t)−ξ(s))^2 = t−s, t>s ∈ R. We consider...

Pages of the article in the issue: 9 - 29Language of the article: Ukrainian

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable $\varphi$-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present th...

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in $L_2(\Omega)$ sense) of the heat equation with random initial conditions given by $\varphi$-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over...

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable φ-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present the exampl...

A model that approximates the fractional Brownian motion with parameter α ∈ (0, 2) with a given reliability 1 − δ, 0<δ<1, and accuracy ε>0inthe space Lp ([0,T]) is constructed. An example of a simulation in the space L2([0, 1]) is given.

In this paper, there are studied properties of a strictly ϕ-sub-Gaussian quasi shot noise process X(t) = integral_{-∞}^{+∞} g(t, u) dξ(u), t ∈ R, generated by the process ξ and the response function g. New estimates for distributions of suprema of such processes are derived. An example of application of the obtained results is given.

In this paper, there are studied properties of stochastic processes belonging to the spaces of φ-sub-Gaussian random variables Sub_φ (Ω). For the processes defined on R, we obtain conditions for boundedness and continuity with probability 1, estimates for the distribution of the supremum are also derived.

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable (Formula presented.)-sub-Gaussian processes. The main results are the bounds for the distributions of...

We consider simulation of '-sub-Gaussian processes that are weakly self-similar with stationary increments in the sense that they have the covariance function R(t,s) = 1 2 t 2H + s 2H | t s| 2H for some H 2 (0,1). Here such processes are referred to as processes of generalized fractional Brownian motion, since the second order structure of the proc...

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable $\varphi$-sub-Gaussian processes. The main results are the bounds for the distributions of the suprem...

In the paper, strictly f-sub-Gaussian quasi shot noise processes are considered. There are obtained estimates for distribution of supremum of such a process defined on a compact set and formulated conditions for its sample functions continuity with probability one.

Summary: The life and main achievements of an outstanding mathematician and talented educator, Doctor of Physics and Mathematics, Professor Mykhailo Iosypovych Yadrenko (16.04.1932 -- 28.09.2004) are described. His personality features are also pointed out.

We study the Lipschitz continuity of generalized sub-Gaussian processes and provide estimates for the distribution of the norms of such processes. The results are applied to the case of weakly self-similar generalized sub-Gaussian processes with stationary increments (the fractional Brownian motion is a particular case of these processes).

Random processes from the class V (ϕ, ψ) which is more general than the class of ψ-sub-Gaussian random process. The upper estimate of the probability that a random process from the class V (ϕ, ψ) exceeds some function is obtained. The results are applied to generalized process of fractional Brownian motion.

In the paper we present conditions for uniform convergence with probability one of wavelet expansions of ℓ-sub-Gaussian (in particular, Gaussian) random processes defined on the space R. It is shown that upon certain conditions for the bases of wavelets the wavelet expansions of stationary almost sure continuous Gaussian processes and wavelet expan...

We consider simulation of \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\)-processes that are weakly selfsimilar with stationary increments in the sense that they have the covariance function $$R{\left( {t,s} \right)} = \frac{1}{2}{\left( {t^{{2H}} + s^{{2H}} - {\left| {t - s} \right|}^{{2H}} } \right)}$$for some H ∈ (0, 1). This means that the...

In this paper we consider random process from the space $\Sub_{\varphi}(\Omega)$, which is defined on compact set, and the probability that supremum of this process exceeds some function. The class of $\Sub_{\varphi}(\Omega)$ random processes is more general than the class of Gaussian processes. By applying obtained estimation to a fluid queue fed...

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having d...

We study the Lipschitz continuity of generalized sub-Gaussian processes, and provide estimates for the distribution of the norms of such processes. The results are applied to the case of weakly self-similar stationary- increment generalized sub-Gaussian processes (the fractional Brownian mo- tions are special cases).

We describe the present situation on teaching Survey Sampling in the Department of Probability Theory and Mathematical Statistics at Kyiv National University. The shortcomings are analyzed and needed changes are suggested.