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The main purpose of this study is the construction of a model of a Gaussian stochastic process with given reliability and accuracy in some Orlicz spaces. In the paper, a suitable model is presented, conditions for the model parameters are derived, and some examples of their calculations are 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 $$\varphi $$ -sub-Gaussian processes. The main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations. The results obtained in the paper hold, in particular, for the case of Gaussian initial condition.

We offer in this paper the non-asymptotical pairwise bilateral exact up to multiplicative constants interrelations between the tail behavior, moments (Grand Lebesgue Spaces) norm and Orlicz’s norm for random variables (r.v.), which does not satisfy in general case the Kramer’s condition.

The expansions with uncorrelated coefficients in function systems generated by wavelets are constructed in the paper for second order stochastic processes. Conditions for the uniform convergence with probability one on a finite interval are found for expansions whose coefficients are independent. Conditions for the uniform convergence in probability on a finite interval are found for expansions of strictly φ-sub-Gaussian stochastic processes.

We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval.

In this paper odd-order heat-type equations with different random initial conditions are examined. In particular, we give rigorous conditions for the existence of the solutions in the case where the initial condition is represented by a strictly ϕ –subGaussian harmonizable process η = η (x). Also the case where η is represented by a stochastic integral with respect to a process with independent increment is studied.

Estimates for the distribution of the supremum of square-Gaussian stochastic processes defined on R+ are found in the paper. Using these results, we find estimates for the deviation in the uniform metric between the correlogram and the correlation function of a real stationary Gaussian stochastic process. A criterion for testing a hypothesis concerning the correlation function is also constructed.

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L p (Ω) and L 2 (Ω) in the norm of the space L q (ℝ).

We study the exponential type inequalities for the distribution of the supremum of a random field that arises as the solution of the heat equation with a random initial condition that is a strictly sub-Gaussian random field.

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 different Hurst indices.

We construct models of Gaussian isotropic random fields on a sphere in an n-dimensional space. The models approximate the fields with a given reliability and accuracy in the space L p , p ≥ 2.

On the basis of the limit theorem for quadratic variation, we construct a consistent estimator for parameters of a homogeneous Gaussian random field of a certain class. We find confidence ellipsoids for estimators of this sort.

We study random processes from quasi-BanachK σ-spaces of random variables whose domain of definition is not necessarily a compact set. We establish conditions under which the supremum of a properly normalized process belongs to the same space as the process itself. We also obtain estimates for the norm of this supremum.

A test is proposed for a hypothesis on the correlation function of general Gaussian random processes. The test is based on theorems on estimates of the distribution of the supremum of sample estimators of correlation functions of Gaussian processes. For a wide class of stationary processes formulas are given that allow the test to be used immediately. Bibliography: 4 titles.

We consider computer-implementable models that approximate Gaussian stochastic processes with given accuracy and reliability in L2(0, T).

Models of stationary Gaussian stochastic processes with discrete and continuous spectra are constructed. Simulation of stationary Gaussian processes with a continuous spectrum is considered for the following cases: when the covariance function of the stochastic process is expandable in a Fourier series with positive coefficients; when the spectrum of the stationary Gaussian stochastic process is concentrated on the interval [0, Λ]; and in the general case. The stationary Gaussian process is simulated with prescribed reliability and accuracy in L2(0, T).

A survey of the works of M. I. Yadrenko in the theory of random fields on finite-dimensional and infinite-dimensional spaces is presented.