Abstract
Consider the random matrix [Formula: see text] where D and [Formula: see text] are deterministic Hermitian nonnegative matrices with respective dimensions N × N and n × 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 [Formula: see text] converge as N, n → ∞ 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 [Formula: see text] near an edge a of its support. In addition, a complete characterization of the support of μ is provided. Aside from its mathematical interest, the analysis underlying these results finds important applications in a certain class of statistical estimation problems.
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