# zbMATH — the first resource for mathematics

Rate optimality of wavelet series approximations of fractional Brownian motion. (English) Zbl 1050.60043
Summary: Consider the fractional Brownian motion process $$B_H (t)$$, $$t\in[0,T]$$, with parameter $$H\in(0,1)$$. Y. Meyer, F. Sellan and M. S. Taqqu [J. Fourier Anal. Appl. 5, No. 5, 465–494 (1999; Zbl 0948.60026)] have developed several random wavelet representations for $$B_H(t)$$, of the form $$\sum^\infty_{k=0} U_k(t)\varepsilon_k$$ where $$\varepsilon_k$$ are Gaussian random varaibles and where the functions $$U_k$$ are not random. Based on the results of T. Kühn and W. Linde [Bernoulli 8, No. 5, 669–696 (2002; Zbl 1012.60074)], we say that the approximation $$\sum^n_{k=0} U_k(t) \varepsilon_k$$ of $$B_H(t)$$ is optimal if $\left(E\sup_{t\in[0,T]} \left |\sum^\infty_{k=n} U_k(t) \varepsilon_k\right |^2\right)^{1/2} =O\bigl(n^{-H} (1+\log n)^{1/2} \bigr)$ as $$n\to\infty$$. We show that the random wavelet representations given by Meyer, Sellan and Taqqu (loc. cit.) are optimal.

##### MSC:
 60G15 Gaussian processes 60G18 Self-similar stochastic processes 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 65T60 Numerical methods for wavelets
Full Text: