Krein's Spectral Theory and the Paley-Wiener Expansion for Fractional Brownian Motion

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein's work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an...

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Bibliographic Details
Published in:The Annals of probability 2005-03, Vol.33 (2), p.620-644
Main Authors: Dzhaparidze, Kacha, van Zanten, Harry
Format: Article
Language:English
Subjects:
60G15
60G51
62M15
Bessel functions
Brownian motion
Exact sciences and technology
Fourier transformations
Fractional Brownian motion
Fractions
Infrared radiation
Inner products
Martingales
Mathematical models
Mathematics
Polynomials
Probability
Probability and statistics
Probability theory and stochastic processes
Random variables
Sciences and techniques of general use
Series expansion
spectral analysis
Spectral theory
Stochastic analysis
Stochastic processes
Studies
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Summary:In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein's work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of the classical Paley-Wiener expansion of the ordinary Brownian motion to the fractional case.
ISSN:0091-1798
2168-894X
DOI:10.1214/009117904000000955