Krein’s spectral theory and the Paley-Wiener expansion for fractional Brownian motion. (English) Zbl 1083.60028

Let \(\{X\},t\geq0\), be a fractional Brownian motion (fBm) with Hurst index \(H\in(0,1)\) and, for \(T>0\), let \(L_T\) be the closed Hilbert space spanned by the functions \(e_t(\lambda)=(e^{it\lambda}-1)/i\lambda\), \(t\in[0,T]\). The authors develop rich spectral theory for fBm. They represent the fBm as integral with respect to a fundamental martingale and obtain an isometry between \(L_T\) and the function space corresponding to the fundamental martingale. The isometry is a Fourier-type transformation whose kernel is given explcitly. The latter is used to turn \(L_T\) into a reproducing kernel Hilbert space. The authors then introduce functions whose role here is analogous to that of orthogonal polynomials in time series. This is used to obtain a closed form expression for the reproducing kernel and an orthonormal basis in \(L_T\).
The spectral theory developed by the authors allows them to generalise the classical Paley-Wiener expansion to the fractional case. They show that the complex fBm admits the series expansion \(\sum_{n} ((e^{2i\omega_nt}-1)/2i\omega_n)Z_n\), \(t\in[0,1],\) where the sum is over all integers, \(\omega_n\) are the real-valued zeroes of the Bessel function \(J_{1-H}\) and \(Z_n\) are independent complex Gaussian random variables. The classical result is indeed a particular case of this since for \(H=1/2\) the variables \(Z_n\) are standard Gaussian and \(\omega_n=n\pi\). The authors consider also the rate of convergence of the Paley-Wiener expansion of fBm and obtain an optimality property for it.


60G15 Gaussian processes
60G35 Signal detection and filtering (aspects of stochastic processes)
60G51 Processes with independent increments; Lévy processes
62M15 Inference from stochastic processes and spectral analysis
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