Spectra of infinite-dimensional sample covariance matrices

Abstract

We study spectral functions of infinite-dimensional random Gram matrices of the form RRT, where R is a rectangular matrix with an infinite number of rows and with the number of columns N → ∞, and the spectral functions of infinite sample covariance matrices calculated for samples of volume N → ∞ under conditions analogous to the Kolmogorov asymptotic conditions. We assume that the traces d of the expectations of these matrices increase with the number N such that the ratio d/N tends to a constant. We find the limiting nonlinear equations relating the spectral functions of random and nonrandom matrices and establish the asymptotic expression for the resolvent of random matrices.