Uniform convergence in probability of wavelet expansions of random processes from L 2(Ω)

Abstract

In the paper we present conditions for uniform convergence in probability on [0, T ] of wavelet expansions of random process X = {X(t), t ∈ R}, with E X(t) = 0, E |X(t)| 2 < ∞. We obtain convergence rate of wavelet representation for random processes in the space C(0, T ) as well.

On convergence of general wavelet decompositions of nonstationary stochastic processes

“…Theorem 1. [18] Let X(t), t ∈ R, be a random process such that EX(t) = 0, E|X(t)| 2 < ∞ for all t ∈ R, and its covariance function R(t, s) := EX(t)X(s) is continuous. Let the fwavelet φ and the m-wavelet ψ be continuous functions which satisfy assumption S(1).…”

“…Recently, a considerable attention was given to the properties of the wavelet orthonormal series representation of random processes. More information on convergence of wavelet expansions of random processes in various spaces, references and numerous applications can be found in [3,7,14,15,16,17,18,21,20,24]. Most known stochastic results concern the mean-square or almost sure convergence, but for various practical applications one needs to require uniform convergence.…”

Uniform Convergence of Wavelet Expansions of Gaussian Random Processes

“…The organization of this article is the following. In the second section we introduce the necessary background from wavelet theory and certain sufficient conditions for mean-square 2 convergence of wavelet expansions in the space L 2 (Ω), obtained in [9]. In §3 we formulate and discuss the main theorem on uniform convergence in probability of the wavelet expansions of stationary Gaussian random processes.…”

“…Recently, a considerable attention was given to the properties of the wavelet transform and of the wavelet orthonormal series representation of random processes. More information on convergence of wavelet expansions of random processes in various spaces, references and numerous applications can be found in [2,5,7,8,9,13,18].…”

Conditions for the uniform convergence in probability of wavelet decompositions for stochastic processes from the space $\operatorname{Exp}_{\varphi}(\Omega)$

“…The uniform convergence of wavelet decompositions of nonrandom functions is considered in the book [6]. Some problems related to the uniform convergence with probability one and in probability of wavelet decompositions of stochastic processes are studied in the papers [7,8,11,12] for various spaces of random variables.…”