In this paper, we are interested in sequences of q-tuple of N × 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 Thorbjørnsen. 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. We mention a couple of applications in random matrix and operator space theory.Dans cet article, nous nous intéressons au q-tuple de matrices N × N qui ont une distribution limite forte (i.e. pour tout polynôme non-commutatif en les matrices et leurs adjoints, sa trace normalisée et sa norme convergent). Nous partons d'une telle suite de matrices aléatoires et montrons que cette propriété persiste si on rajoute au q-tuple des matrices indépandantes unitaires distribudes suivant la mesure de Haar. Par ailleurs, la limite des normes et des traces en des polynômes non-commutatifs en la suite élargie peut être calculé avec la construction du produit libre réduit. Ceci étend les résultats d'un des auteurs (C.M.) et de Haagerup et Thorbjørnsen. Nous montrons aussi qu'un p-tuple de matrices indépandantes orthogonales et symplectiques ont une distribution limite forte, étendant par là-même un résultat de Schultz. Nous passons aussi en revue quelques applications de notre résultat aux matrices aléatoires et à la théorie des espaces d'opórateur.

abstract:Let X N = (X (N ) 1 , . . . , X (N ) p ) be a family of N ×N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices), possibly random but independent of X N , for which the operator norm of P (X N , Y N , Y * N ) converges almost surely for all polynomials P . Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y N and of the polynomials P , we get for a large class of matrices the "no eigenvalues outside a neighborhood of the limiting spectrum" phenomena. We give examples of diagonal matrices Y N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.

Abstract. We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.