On bilinear forms based on the resolvent of large random matrices - Archive ouverte HAL Access content directly
Journal Articles Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Year : 2013

## On bilinear forms based on the resolvent of large random matrices

Walid Hachem
Philippe Loubaton
Jamal Najim
• Function : Author
• PersonId : 863742
Pascal Vallet

#### Abstract

Consider a matrix $\Sigma_n$ with random independent entries, each non-centered with a separable variance profile. In this article, we study the limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$ and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to $\Sigma_n \Sigma_n^*$ as the dimensions of matrix $\Sigma_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $\Sigma_n \Sigma_n^*$ which do not only depend on the eigenvalues of $\Sigma_n \Sigma_n^*$, 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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-00474126 , version 1 (19-04-2010)
hal-00474126 , version 2 (23-08-2011)

### Identifiers

• HAL Id : hal-00474126 , version 2
• ARXIV :
• DOI :

### Cite

Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet. On bilinear forms based on the resolvent of large random matrices. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2013, 49 (1), pp.36-63. ⟨10.1214/11-AIHP450⟩. ⟨hal-00474126v2⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

186 View