Abstract. Voiculescu's asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gaussian matrices) by applying the first result to the unitary parts of their diagonalization. Bi-unitarily invariant non-selfadjoint random matrices are also treated via polar decomposition.

AMS Subject Classification (1991): 15A52, 62E20, 60F99

Keyword(s): random matrices, free probability, asymptotic freeness

Received January 14, 2000. (Registered under 2769/2009.)