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

Abstract

Let X n 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 R n an n × N random matrix independent of X n . Assume, almost surely, as n → ∞ , the empirical distribution function (e.d.f.) of the eigenvalues of 1 N R 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 n + σ X n ) ( R n + σ X n ) * converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.