Consider the random matrix Σ=D1/2XD1/2 where D
and D 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 D 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
∣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.Comment: Correction of the proof of Lemma 3.