Asymptotic study of covariance operators of fractional processes: analytic approach with applications

Eigenproblems frequently arise in theory and applications of stochastic processes, but only a few have explicit solutions. Those which do, are usually solved by reduction to the generalized Sturm-Liouville theory for differential operators. This includes the Brownian motion and a whole class of processes, which derive from it by means of linear transformations. The more general eigenproblems are not solvable in closed form and the subject of my talk is the asymptotic spectral analysis of the fractional Gaussian processes. I will present a new methodology for the spectral analysis of the fractional type covariance operators, that allows to find accurate second order asymptotic approximations for both the eigenvalues and the eigenfunctions, and some applications of the developed theory.