Empirical distribution of eigenvalues of covariance matrices of ar processes. A simulation study

Overview & Synopsis

We consider a class of random matrices Bn = X^Tj^Xj^, where X/v is a (N x n(N)) matrix consisting of indépendant rows, each row contains entries Xij satisfying an autoregressive relation AR(1) and Tjv is a diagonal matrix independent of X/v-Following the paper of Silvertein and Bai [8], we derived in [6] that if the eigenvalues empirical distribution function of converges almost surely to a proper probability distribution as N-> oo and jj-» c > 0, then the eigenvalues empirical distribution function of Bjsr converges almost surely to a nonrandom limit given by Marcenko and Pastur in [7]. We essentially focalize on behavior of empirical estimators of densities of eigenvalues distribution function of Bat, illustrating this by numerical simulations. Selecting differents values of the dimension c, AR parameter p, size sample N and the eigenvalues of Tjv, we analyze via Ti-error their effects on the performance and convergence rate of estimators. Keywords. Large dimensional random matrices. Engenvalues empirical distribution function. Density function estimator. Autoregressive Processes. Stieltjes Transform.