Existence and Analysis of the Limiting Spectral Distribution of Large Dimensional Information-Plus-Noise Type Matrices

Abstract

Let X[subscript n] be n by N with i.i.d. complex entries having unit variance (sum of variances of real and imaginary parts equals 1), s>0 constant, and R[subscript n] an n by N random matrix independent of X[subscript n]. Assume, almost surely, as n goes to infinity, the empirical distribution function (e.d.f.) of the eigenvalues of (1/N)R[subscript n]R*n converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio n/N tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of (1/N)(R[subscript n] + sX[subscript n])(R[subscript n] + sX[subscript n])* converges in distribution to a nonrandom p.d.f. being characterized in terms of its Stieltjes transform, which satisfies a certain equation. It is also shown that, away from zero, the limiting distribution possesses a continuous density. The density is analytic where it is positive and, for the most relevant cases of a in the boundary of its support, exhibits behavior closely resembling that of the square root of |x-a| for x near a. A procedure to determine its support is also analyzed.