On bilinear forms based on the resolvent of large random matrices

Hachem, Walid; Loubaton, Philippe; Najim, Jamal; Vallet, Pascal

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 36-63

Consider a $N\times n$ non-centered matrix ${𝛴}_n$ with a separable variance profile:

$${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$

Matrices $D_n$ and ${\tilde{D}}_n$ are non-negative deterministic diagonal, while matrix $A_n$ is deterministic, and $X_n$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_n\left(z\right)$ the resolvent associated to ${𝛴}_n{𝛴}_n^{*}$, i.e.

$$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$

Given two sequences of deterministic vectors $\left(u_n\right)$ and $\left(v_n\right)$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:

$$u_n^{*}{Q}_n\left(z\right)v_n\forall z\in ℂ-{ℝ}^{+},$$

as the dimensions of matrix ${𝛴}_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of ${𝛴}_n{𝛴}_n^{*}$ which do not only depend on the eigenvalues of ${𝛴}_n{𝛴}_n^{*}$, 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.

Considérons une matrice ${𝛴}_n$, non centrée, de taille $N\times n$, avec un profil de variance séparable :

$${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$

Les matrices $D_n$ et ${\tilde{D}}_n$ sont déterministes, diagonales et non négatives ; la matrice $A_n$ est déterministe ; la matrice $X_n$ 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 $Q_n\left(z\right)$ la résolvante associée à ${𝛴}_n{𝛴}_n^{*}$, i.e.

$$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$

Étant données deux suites déterministes de vecteurs $\left(u_n\right)$ et $\left(v_n\right)$ de norme euclidienne bornée, on étudie le comportement asymptotique de la forme bilinéaire aléatoire :

$$u_n^{*}{Q}_n\left(z\right)v_n\forall z\in ℂ-{ℝ}^{+},$$

quand les dimensions de la matrice ${𝛴}_n$ tendent vers l’infini au même rythme. De telles quantités apparaissent dans l’étude de fonctionnelles de ${𝛴}_n{𝛴}_n^{*}$ ne dépendant pas uniquement des valeurs propres de ${𝛴}_n{𝛴}_n^{*}$, 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.

Classification : Primary 15A52, secondary, 15A18, 60F15
Keywords: random matrix, empirical distribution of the eigenvalues, Stieltjes transform

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