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

2013/10/30 by Couillet, Romain, Hachem, Walid · 3 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1310.8094

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 × 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 \widetilde 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.

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