On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices

Let Xn be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ>0 constant, and Rn an n×N random matrix independent of Xn. Assume, almost surely, as n→∞, the empirical distribution function (e.d.f.) of the eigenvalues of converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.