Etude Spectrale de Matrices de Covariance d′un Processus Autorégressif AR.

Abstract

This thesis deals with the study of the limit spectral distribution of a class of large random matrices having correlated entries. We are interested in the asymptotic behavior of large covariance matrices whose entries are correlated by the relation of an autoregressive of order one. In this context, we show that the empirical eigenvalue distribution function of the covariance matrix converges almost surely to a non-random function given by Marcenko and Pastur. Our approach consists of a centralization and a truncation of strongly geometrically mixing random variables and an application of the Stieltjes transform method which makes it possible to obtain, under certain conditions, the integral equation of the limit spectral distribution, and we thus show a universality result concerning the asymptotic behavior of the spectrum of these covariance matrices. We also investigate the case of matrices whose columns are autoregressive processes of order one. This extends on the one hand the Örst result to a vector framework and on the other hand generalizes the results on matrices with independent identically distributed entries. Finally, we present numerical simulations illustrating the behavior of the estimator of the spectral density of the matrices in question around the true density by varying the various parameters.