  In this paper, we are interested in sequences of q-tuple of N-by-N random
matrices having a strong limiting distribution (i.e. given any non-commutative
polynomial in the matrices and their conjugate transpose, its normalized trace
and its norm converge). We start with such a sequence having this property, and
we show that this property pertains if the q-tuple is enlarged with independent
unitary Haar distributed random matrices. Besides, the limit of norms and
traces in non-commutative polynomials in the enlarged family can be computed
with reduced free product construction. This extends results of one author (C.
M.) and of Haagerup and Thorbjornsen. We also show that a p-tuple of
independent orthogonal and symplectic Haar matrices have a strong limiting
distribution, extending a recent result of Schultz.

Operator Algebras;Probability