The Empirical Distribution of the Eigenvalues of a Gram Matrix with a given Variance Profile

Consider a N × n random matrix Yn = (Y n ij ) where the entries are given by Y n ij = σ(i/N,j/n) √ n X ij , the X n ij being centered i.i.d. and σ : [0, 1] 2 → (0,∞) being a continuous function called a variance profile. Consider now a deterministic N×n matrix Λn = ( Λij ) whose non diagonal elements are zero. Denote by Σn the non-centered matrix Yn+Λn. Then under the assumption that limn→∞ N n = c > 0 and 1 N N ∑ i=1 δ i N ,(Λnii) 2 ) −−−−→ n→∞ H(dx, dλ), where H is a probability measure, it is proven that the empirical distribution of the eigenvalues of ΣnΣ T n converges almost surely in distribution to a non random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication. Résumé. Soit Yn = (Y n ij ) une matrice N × n dont les entrées sont données par Y n ij = σ(i/N,j/n) √ n X ij , les X n ij étant des variables aléatoires centrées, i.i.d. et où σ : [0, 1] → (0,∞) est une fonction continue qu’on appellera profil de variance. Considérons une matrice déterministe Λn = ( Λij ) de dimensions N × n dont les éléments non diagonaux sont nuls. Appelons Σn la matrice non centrée définie par Σn = Yn + Λn. Sous les hypothèses que limn→∞ N n = c > 0 et que