Limit distributions for functionals of Gaussian processes

Abstract

This thesis is devoted to the study of the convergence in distribution of functionals of Gaussian processes. Most of the problems that we present are addressed by using an approach based on Malliavin calculus techniques. Our main contributions are the following: First, we study the asymptotic law of the approximate derivative of the self-intersection local time (SILT) in $[0,T]$ for the fractional Brownian motion. In order to do this, we describe the asymptotic behavior of the associated chaotic components and show that the first chaos approximates the SILT in $L^2$. Secondly, we examine the asymptotic law of the approximate self-intersection local time process for the fractional Brownian motion. We achieve this in two steps: the first part consists on proving the convergence of the finite dimensional distributions by using the `multidimensional fourth moment theorem'. The second part consists on proving the tightness property, for which we follow an approach based on Malliavin calculus techniques. The third problem consists on proving a non-central limit theorem for the process of weak symmetric Riemann sums for a wide variety of self-similar Gaussian processes. We address this problem by using the so-called small blocks-big blocks methodology and a central limit theorem for the power variations of self-similar Gaussian processes. Finally, we address the problem of determining conditions under which the eigenvalues of an Hermitian matrix-valued Gaussian process collide with positive probability.