On uniform convergence of wavelet expansions of φ-sub-Gaussian random processes

Abstract

In the paper we present conditions for uniform convergence with probability one of wavelet expansions of ϕ-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 expansions of fractional Brownian motion converge uniformly with probability one on any finite interval.

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

“…This lemma is a simple consequence of Lemma 2.1 and Lemma 2.3 from the paper [8]. This lemma is a simple consequence of Lemma 2.1 and Lemma 2.3 from the paper [8].…”

“…That is, we find the estimates for Conditions of convergence of wavelet expansions in different spaces were considered in [6] and conditions of uniform convergence with probability one in [8]. That is, we find the estimates for Conditions of convergence of wavelet expansions in different spaces were considered in [6] and conditions of uniform convergence with probability one in [8].…”

Uniform Convergence of Wavelet Expansions of 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].…”

“…where k n := (k 0 , ..., k n−1 ). Contrary to many theoretical results (see, for example, [8,13]) with infinite series form of X n,kn (t), in direct numerical implementations we always consider truncated series like (1), where the number of terms in the sums is finite by application reasons. However, there are almost no stochastic results on uniform convergence of finite wavelet expansions to X(t).…”

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.…”