Krein's spectral theory and the PaleyWiener expansion for fractional Brownian motion
Abstract
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 PaleyWiener expansion of the ordinary Brownian motion to the fractional case.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2005
 arXiv:
 arXiv:math/0503656
 Bibcode:
 2005math......3656D
 Keywords:

 Mathematics  Probability;
 60G15;
 60G51;
 62M15 (Primary)
 EPrint:
 Published at http://dx.doi.org/10.1214/009117904000000955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)