## 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.

### MSC:

 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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### References:

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