NO EIGENVALUES OUTSIDE THE SUPPORT OF THE LIMITING SPECTRAL DISTRIBUTION OF INFORMATION-PLUS-NOISE TYPE MATRICES

Abstract

We consider a class of matrices of the form Cn = (1/N)(Rn + σXn)(Rn + σXn)*, where Xn is an n × N matrix consisting of independent standardized complex entries, Rn is an n × N nonrandom matrix, and σ > 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is contained in [Formula: see text], but each column of Rn is contaminated by noise. As n → ∞, if n/N → c > 0, and the empirical distribution of the eigenvalues of [Formula: see text] converge to a proper probability distribution, then the empirical distribution of the eigenvalues of Cn converges a.s. to a nonrandom limit. In this paper we show that, under certain conditions on Rn, for any closed interval in ℝ+ outside the support of the limiting distribution, then, almost surely, no eigenvalues of Cn will appear in this interval for all n large.