Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices

Let X be n × N containing i.i.d. complex entries with E |X11 − EX11|2 = 1, and T an n × n random Hermitian nonnegative definite, independent of X. Assume, almost surely, as n → ∞, the empirical distribution function (e.d.f.) of the eigenvalues of T converges in distribution, 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) XX*T converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.