Abstract:
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.
Keywords:spectra of random matrices, spectral functions of sample covariance matrices, spectra of infinite-dimensional random matrices.
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