Dedicated to the Sixtieth Birthday of Min-Te Chao Let Bn =(1/N)T 1/2

n where Xn is n × N with i.i.d. complex standardized entries having finite fourth moment, and T 1/2 n is a Hermitian square root of the nonnegative definite Hermitian matrix Tn. It is known that, as n →∞,ifn/N converges to a positive number, and the empirical distribution of the eigenvalues of Tn converges to a proper probability distribution, then the empirical distribution of the eigenvalues of Bn converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of Tn, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all n sufficiently large