Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices

Let X be n x N containing i.i.d. complex entries with E \X(11) - EX(11)\(2)=1, and T an n x n random Hermitian nonnegative definite, independent ofX. Assume, almost surely, as n --> infinity, the empirical distribution function (e.d.E) of the eigenvalues or. 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. (C) 1995 Academic Press, Inc.