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 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.Comment: 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

Representations of isotropic random fields with homogeneous increments, with applications to spacial fractional Brownian motion

This is a brief account of the current work by Dzhaparidze, van Zanten and Zareba, delivered as a lecture note at the conference Small deviations and related topics II held in St. Petersburg, September 12-19, 200