The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?Q%20=%20%5Cfrac%7B1%7Dn%20X_1X%5E%5Cprime_1%20+%20%5Ccdot%5Ccdot%5Ccdot%20+%20%5Cfrac%7B1%7Dn%20X_kX%5E%5Cprime_k," />  where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?Q%20=%20%5Cfrac%7B1%7Dn%20X_1X%5E%5Cprime_1%20+%20%5Ccdot%5Ccdot%5Ccdot%20+%20%5Cfrac%7B1%7Dn%20X_kX%5E%5Cprime_k," />  where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

Cette thèse s'inscrit dans la théorie des matrices aléatoires, à l'intersection avec la théorie des probabilités libres et des algèbres d'opérateurs. Elle s'insère dans une démarche générale qui a fait ses preuves ces dernières décennies : importer les techniques et les concepts de la théorie des probabilités non commutatives pour l'étude du spectre de grandes matrices aléatoires. On s'intéresse ici à des généralisations du théorème de liberté asymptotique de Voiculescu. Dans les Chapitres 1 et 2, nous montrons des résultats de liberté asymptotique forte pour des matrices gaussiennes, unitaires aléatoires et déterministes. Dans les Chapitres 3 et 4, nous introduisons la notion de fausse liberté asymptotique pour des matrices déterministes et certaines matrices hermitiennes à entrées sous diagonales indépendantes, interpolant les modèles de matrices de Wigner et de Lévy. / The thesis fits into the random matrix theory, in intersection with free probability and operator algebra. It is part of a general approach which is common since the last decades: using tools and concepts of non commutative probability in order to get general results about the spectrum of large random matrices. Where are interested here in generalization of Voiculescu's asymptotic freeness theorem. In Chapter 1 and 2, we show some results of strong asymptotic freeness for gaussian, random unitary and deterministic matrices. In Chapter 3 and 4, we introduce the notion of asymptotic false freeness for deterministic matrices and certain random matrices, Hermitian with independent sub-diagonal entries, interpolating Wigner and Lévy models.