No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Information-Plus-Noise Type Matrices

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, Rj 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 (1/N)RnR ∗ n, but each column of Rn is contaminated by noise. As n → ∞, if n/N → c > 0, and the empirical distribution of the eigenvalues of (1/N)RnRn 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.