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Wavelet analysis and synthesis of fractional Brownian motion. (English) Zbl 0743.60078
Summary: Fractional Brownian motion (fBm) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and $1/f$- type spectral behavior over wide ranges of frequencies. Statistical self- similarity is an essential feature of fBm and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of fBm. It reveals a stationary structure at each scale and a power-law behavior of the coefficients’ variance from which the fractal dimension of fBm can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered and, finally, some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out.

##### MSC:
 60J65 Brownian motion 60G15 Gaussian processes