On bilinear forms based on the resolvent of large random matrices
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 36-63.

Considérons une matrice 𝛴n, non centrée, de taille N×n, avec un profil de variance séparable :

𝛴n=Dn1/2XnD˜n1/2n+An.
Les matrices Dn et D˜n sont déterministes, diagonales et non négatives ; la matrice An est déterministe ; la matrice Xn est une matrice aléatoire dont les entrées complexes sont des variables aléatoires indépendantes et identiquement distribuées, de moyenne nulle et de variance unité. On note Qn(z) la résolvante associée à 𝛴n𝛴n*, i.e.
Qn(z)=𝛴n𝛴n*-zIN-1.
Étant données deux suites déterministes de vecteurs (un) et (vn) de norme euclidienne bornée, on étudie le comportement asymptotique de la forme bilinéaire aléatoire :
un*Qn(z)vnz-+,
quand les dimensions de la matrice 𝛴n tendent vers l’infini au même rythme. De telles quantités apparaissent dans l’étude de fonctionnelles de 𝛴n𝛴n* ne dépendant pas uniquement des valeurs propres de 𝛴n𝛴n*, et sont centrales dans l’étude de problèmes relatifs aux matrices de Gram non centrées tels que l’établissement de théorèmes de la limite centrale, le comportement des entrées individuelles et les problèmes de séparation des valeurs propres.

Consider a N×n non-centered matrix 𝛴n with a separable variance profile:

𝛴n=Dn1/2XnD˜n1/2n+An.
Matrices Dn and D˜n are non-negative deterministic diagonal, while matrix An is deterministic, and Xn is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by Qn(z) the resolvent associated to 𝛴n𝛴n*, i.e.
Qn(z)=𝛴n𝛴n*-zIN-1.
Given two sequences of deterministic vectors (un) and (vn) with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:
un*Qn(z)vnz-+,
as the dimensions of matrix 𝛴n go to infinity at the same pace. Such quantities arise in the study of functionals of 𝛴n𝛴n* which do not only depend on the eigenvalues of 𝛴n𝛴n*, and are pivotal in the study of problems related to non-centered Gram matrices such as central limit theorems, individual entries of the resolvent, and eigenvalue separation.

DOI : 10.1214/11-AIHP450
Classification : Primary 15A52, secondary, 15A18, 60F15
Mots-clés : random matrix, empirical distribution of the eigenvalues, Stieltjes transform
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     title = {On bilinear forms based on the resolvent of large random matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {36--63},
     publisher = {Gauthier-Villars},
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     mrnumber = {3060147},
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Hachem, Walid; Loubaton, Philippe; Najim, Jamal; Vallet, Pascal. On bilinear forms based on the resolvent of large random matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 36-63. doi : 10.1214/11-AIHP450. https://www.numdam.org/articles/10.1214/11-AIHP450/

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