Abstract
The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed technique are shown for several classes of stochastic processes. In particular, the main theorem is adjusted to the fractional Brownian motion case. New results on the rate of convergence of the wavelet expansions in the space $C([0,T])$ are also presented.
Highlights
In the book [11] wavelet expansions of non-random functions bounded on R were studied in different spaces
We clearly see that empirical probabilities of obtaining large reconstruction errors become smaller if the number of terms in the wavelet expansions increases
We prove the exponential rate of convergence of the wavelet expansions
Summary
In the book [11] wavelet expansions of non-random functions bounded on R were studied in different spaces. In the majority of cases, which are interesting from theoretical and practical application points of view, stochastic processes have almost surely unbounded sample paths on R. We clearly see that empirical probabilities of obtaining large reconstruction errors become smaller if the number of terms in the wavelet expansions increases This effect is expected, it has to be established theoretically in a stringent way for different classes of stochastic processes and wavelet bases. The conditions are weaker than those in the former literature These are novel results on stochastic uniform convergence of general finite wavelet expansions of nonstationary random processes. It should be mentioned that the analysis of the rate of convergence gives a constructive algorithm for determining the number of terms in the wavelet expansions to ensure the uniform approximation of stochastic processes with given accuracy. The same symbol C may be used for different constants appearing in the same proof
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