Identiication of Filtered White Noises and of Elliptic Gaussian Random Processes

In this paper , two classes of Gaussian Processes having locally the same fractal properties as Fractional Brownian Motion are studied. Our aim is to give estimators of the relevant parameters of these processes from one sample path. In the rst class (i.e. Filtered White Noises), a time dependency of the integrand of the classical Wiener Integral associated to the Fractional Brownian Motion is introduced. We show how to identify the asymptotic expansion for high frequencies of these integrands on one sample path. Then the identiication of the rst terms of this expansion is used to solve some \\ltering" problems. Furthermore rates of convergence of the estimators are then given. In the second class (i.e. Elliptic Gaussian Random Processes), spatial modulations in the symbol of the pseudo-diierential operator characterizing the reproducing spaces of the processes is introduced. At last identiication results for the symbol of the Elliptic Gaussian Random Processes are deduced from the identiication results for Filtered White Noises.