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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 a time-invariant continuous linear system with a real-valued impulse response function is considered. A new method for the estimator construction of the impulse response function is proposed. Two criteria on the shape of the impulse response function are given. In this paper a time-invariantcontinuous linear system with a real-valued impulse response function is considered. A new method for the estimator construction of the impulse response function is proposed. Two criteria on the shape of the impulse response function are given.

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.

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.

Conditions for justification of the Fourier method for parabolic equations with random initial conditions from Orlicz spaces of random variables are obtained. Bounds for the distribution of the supremum of solutions of such equations are found. We study conditions justifying the application of the Fourier method for parabolic equations with random initial conditions and obtain bounds for the distribution of the supremum of solutions of these equations. Similar problems for hyperbolic equations are considered in [1, 2]. A survey of the corresponding results can be found in [3, 4]. In what follows we consider random initial conditions from the Orlicz spaces of random variables. The paper is organized as follows. Section 1 contains necessary definitions and results of the theory of the Orlicz space. The setting of the problem as well as statements of the main results of the paper is given in Section 2. Conditions for the convergence of stochastic processes in C(T ) and bounds for the distribution of the supremum of solutions of the corresponding equations are presented in Section 3. The proofs of the main results are placed in Section 4. 1. Stochastic processes belonging to an Orlicz space Definition 1.1 ([3]). An even, continuous, convex function U(x) such that U(x) > 0 for x = 0 is called a C-function. Let {Ω, ,P} be a standard probability space. Definition 1.2 ([4]). The space LU (Ω) of random variables ξ(ω) = ξ, ω ∈ Ω, is called the Orlicz space generated by a C-function U(x) if, for any ξ ∈ LU (Ω), there exists a constant rξ such that EU(ξ/rξ) 0: EU ( ξ r ) ≤ 1 } . Definition 1.3. Let X(t) = {X(t), t ∈ T} be a stochastic process. We say that X belongs to the Orlicz space LU (Ω) if, for all t ∈ T , the random variable X(t) belongs to the space LU (Ω). Lemma 1.1 ([3]). Let ξ ∈ LU (Ω) and let EU(ξ/r) ≤ a for some r > 0 and a > 0. Then ‖ξ‖LU ≤ rmax{0; a}. 2000 Mathematics Subject Classification. Primary 60G60, 60G17.