Eigenvalues Distribution Limit of Covariance Matrices with AR Processes Entries

Abstract

We consider a class of random matrices $B_{N}=X_{N}T_{N}X_{N}^{t},$ where $X_{N}$ is a matrix $(N\times n(N))$ whose rows are independent, the entries $X_{ij}$ in each row satisfy an autoregressive relation AR(1), and $T_{N}$ is a diagonal matrix independent of $X_{N}$. Under some conditions, we show that if the empirical distribution function of eigenvalues of $T_{N}$ converges almost surely to a proper probability distribution as $N\longrightarrow \infty$ and $\frac{n(N)}{N}\longrightarrow c>0$, then the empirical distribution function of eigenvalues of $B_{N}$ converges almost surely to a non-random limit function given by Marcenko and Pastur. Numerical simulations illustrate the behavior of kernel density estimators and density estimators of Stieltjes transform around the true density and we give a numerical comparison on the base of $L_{1}$ error varying different parameters.