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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
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