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Linear estimation of self-similar processes via Lamperti’s transformation. (English) Zbl 0963.60034

Summary: Lamperti’s transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

MSC:

60G18 Self-similar stochastic processes
62M20 Inference from stochastic processes and prediction
60G25 Prediction theory (aspects of stochastic processes)
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