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Linear estimation of self-similar processes via Lamperti's transformation

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Carl J. Nuzman  and
H. Vincent Poor
Carl J. Nuzman*
Affiliation:
Princeton University
H. Vincent Poor*
Affiliation:
Princeton University
*
∗Postal address: Dept. of Electrical Engineering, Engineering Quadrangle, Princeton, NJ 08544 5263, USA
∗Postal address: Dept. of Electrical Engineering, Engineering Quadrangle, Princeton, NJ 08544 5263, USA
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Abstract

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.

Keywords

Self-similar processesfractional Brownian motionlinear predictionwhiteninginnovationsscale-stationary processesLamperti's transformationWiener-Kolmogorov filter

MSC classification

Primary: 60G18: Self-similar processes
Secondary: 62M20: Prediction 60G25: Prediction theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 429 - 452
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

Research supported in part by the US Office of Naval Research underGrant N00014–00–1–0141, and in part by the US Department of Defense NDSEG Fellowship Program.

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