Spectra of infinite-dimensional sample covariance matrices

We study spectral functions of infinite-dimensional random Gram matrices of the form $RR^{\mathrm{T}}$, where $R$ is a rectangular matrix with an infinite number of rows and with the number of columns $N\to\infty$, and the spectral functions of infinite sample covariance matrices calculated for samples of volume $N\to\infty$ 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.