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