On an expansion of random processes in series

Abstract

A paper is devoted to new expansions of random processes in the form of series. In particular case the expansions in series of stationary stochastic processes with absolutely continuous spectral function and the expansions with respect to some functions which generate wavelet basis are obtained. These results are used for model construction of stochastic processes in such way that the model approximates the process with given reliability and accuracy in some Banach spaces. The conditions of uniform convergence of Gaussian random series with independent summands are also given.

Conditions for the uniform convergence of expansions of $\varphi $-sub-Gaussian stochastic processes in function systems generated by wavelets

“…Theorem 3.1 is proved in [12] without these restrictions. (It also follows from [13].) We prove Theorem 3.1 below together with Remark 3.1.…”

Simulation of Gaussian Stationary Quasi Ornstein– Uhlenbeck Process with Given Reliability and Accuracy in Spaces C9[0,T]) and Lp([0, T])

“…In the paper we apply representations of random processes in the form of random series with uncorrelated members, obtained in the work by Kozachenko, Rozora, Turchyn (2007), for the construction of models of stochastic processes, which approximates the processes with given reliability and accuracy in spaces  …”

Modeling of Stochastic Processes in L_p(T) Using Orthogonal Polynomials

“…Theorem 1 [1] (On decomposition of the stochastic process using an orthonormal basis) Let X(t), t ∈ T be stochastic process of the second order, EX(t) = 0 ∀t ∈ T , let B(t, s) = EX(t)X(s) be the correlation function of X, let f (t, λ) be some function from L 2 (Λ, µ) space, and let {g k (λ), k ∈ Z} be the orthonormal basis in L 2 (Λ, µ) space. Then, correlation function B(t, s) is represented in the form…”

“…Let us now introduce the model of such process. Definition 1 Let stochastic process X = {X(t), t ∈ T } allow decomposition (1). We will call stochastic process…”

“…let the process X(t) allow representation in the form (1), and the seriesĤ k (t) of Hermite functions is the basis. Let τ φ (ξ k ) = τ ω k , |ω| < 1, and…”