| Let Xn be n x N with i.i.d. complex entries having unit variance (sum of variances of real and imaginary parts equals 1), sigma > 0 constant, and Rn an n x N random matrix independent of Xn. Assume, almost surely, as n → infinity, the empirical distribution function (e.d.f.) of the eigenvalues of 1NRnR*n converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio nN tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of 1N (Rn + sigmaXn)( Rn + sigmaXn)* 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 x-a for x near a. A procedure to determine its support is also analyzed. |
