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Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion. (English) Zbl 0948.60026

Authors’ abstract: We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy’s midpoint displacement technique which is used to generate Brownian motion. The low frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous-time processes whose second-order properties are the same as those of fractional Brownian motion.

MSC:

60G18 Self-similar stochastic processes
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
60F15 Strong limit theorems
60J65 Brownian motion

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References:

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