Analysis of the limiting spectral measure of large random matrices of the separable covariance type

Abstract

Consider the random matrix $Σ= D^{1/2} X \widetilde D^{1/2}$ where $D$ and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \times N$ and $n \times n$, and where $X$ is a random matrix with independent and identically distributed centered elements with variance $1/n$. Assume that the dimensions $N$ and $n$ grow to infinity at the same pace, and that the spectral measures of $D$ and $\widetilde D$ converge as $N,n \to\infty$ towards two probability measures. Then it is known that the spectral measure of $ΣΣ^*$ converges towards a probability measure $μ$ characterized by its Stieltjes Transform. In this paper, it is shown that $μ$ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as $\sqrt{|x - a|}$ near an edge $a$ of its support. A complete characterization of the support of $μ$ is also provided. \\ Beside its mathematical interest, this analysis finds applications in a certain class of statistical estimation problems.