Uniform Convergence of Wavelet Expansions of Gaussian Random Processes

Abstract

New results on uniform convergence in probability for the most general classes of wavelet expansions of stationary Gaussian random processes are given.

Uniform Convergence of Compactly Supported Wavelet Expansions of Gaussian Random Processes

“…In this section we provide examples of wavelets and stationary stochastic processes which satisfy assumption (7) of Theorem 4. Note that the assumptions are simpler than those used in results for stationary processes in [11].…”

“…It imposes minimal additional conditions on wavelet bases, which can be easily verified. The assumptions are weaker than those in the former literature, compare [11,12,14,15]. It is easy to verify that numerous wavelets satisfy these conditions, for example, the well known Daubechies, symmlet, and coiflet wavelet bases, see [9].…”

“…In the recent literature a considerable attention was given to wavelet orthonormal series representations of stochastic processes. Numerous results, applications, and references on convergence of wavelet expansions of random processes in various spaces can be found in [4,7,10,11,12,14,17]. While most known results concern the mean-square convergence, for various practical applications one needs to require uniform convergence.…”

“…[11] Let T = [0, T ], ρ(t, s) = |t−s|. Let X n (t), t ∈ T, be a sequence of Gaussian stochastic processes such that all X n (t) are separable in (T, ρ) and sup n≥1 sup |t−s|≤h…”

On convergence of general wavelet decompositions of nonstationary stochastic processes

“…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.…”