On bilinear forms based on the resolvent of large random matrices

Walid Hachem; Philippe Loubaton; Jamal Najim; Pascal Vallet

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 1, page 36-63
  • ISSN: 0246-0203

Abstract

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Consider a N × n non-centered matrix 𝛴 n with a separable variance profile: 𝛴 n = D n 1 / 2 X n D ˜ n 1 / 2 n + A n . Matrices D n and 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 ( z ) the resolvent associated to 𝛴 n 𝛴 n * , i.e. Q n ( z ) = 𝛴 n 𝛴 n * - z I N - 1 . Given two sequences of deterministic vectors ( u n ) and ( v n ) with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: u n * Q n ( z ) v n ∀ z ∈ 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.

How to cite

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Hachem, Walid, et al. "On bilinear forms based on the resolvent of large random matrices." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 36-63. <http://eudml.org/doc/272084>.

@article{Hachem2013,
abstract = {Consider a $N\times n$ non-centered matrix $\varSigma _\{n\}$ with a separable variance profile: \[\varSigma \_\{n\}=\frac\{D\_\{n\}^\{1/2\}X\_\{n\}\tilde\{D\}\_\{n\}^\{1/2\}\}\{\sqrt\{n\}\}+A\_\{n\}.\] 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 $Q_\{n\}(z)$ the resolvent associated to $\varSigma _\{n\}\varSigma _\{n\}^\{*\}$, i.e. \[Q\_\{n\}(z)=\bigl (\varSigma \_\{n\}\varSigma \_\{n\}^\{*\}-zI\_\{N\}\bigr )^\{-1\}.\] Given two sequences of deterministic vectors $(u_\{n\})$ and $(v_\{n\})$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: \[u\_\{n\}^\{*\}Q\_\{n\}(z)v\_\{n\}\quad \forall z\in \mathbb \{C\} -\mathbb \{R\} ^\{+\},\] as the dimensions of matrix $\varSigma _\{n\}$ go to infinity at the same pace. Such quantities arise in the study of functionals of $\varSigma _\{n\}\varSigma _\{n\}^\{*\}$ which do not only depend on the eigenvalues of $\varSigma _\{n\}\varSigma _\{n\}^\{*\}$, 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.},
author = {Hachem, Walid, Loubaton, Philippe, Najim, Jamal, Vallet, Pascal},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random matrix; empirical distribution of the eigenvalues; Stieltjes transform; bilinear forms; spectral norm; wireless communication system; signal processing},
language = {eng},
number = {1},
pages = {36-63},
publisher = {Gauthier-Villars},
title = {On bilinear forms based on the resolvent of large random matrices},
url = {http://eudml.org/doc/272084},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Hachem, Walid
AU - Loubaton, Philippe
AU - Najim, Jamal
AU - Vallet, Pascal
TI - On bilinear forms based on the resolvent of large random matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 36
EP - 63
AB - Consider a $N\times n$ non-centered matrix $\varSigma _{n}$ with a separable variance profile: \[\varSigma _{n}=\frac{D_{n}^{1/2}X_{n}\tilde{D}_{n}^{1/2}}{\sqrt{n}}+A_{n}.\] 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 $Q_{n}(z)$ the resolvent associated to $\varSigma _{n}\varSigma _{n}^{*}$, i.e. \[Q_{n}(z)=\bigl (\varSigma _{n}\varSigma _{n}^{*}-zI_{N}\bigr )^{-1}.\] Given two sequences of deterministic vectors $(u_{n})$ and $(v_{n})$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: \[u_{n}^{*}Q_{n}(z)v_{n}\quad \forall z\in \mathbb {C} -\mathbb {R} ^{+},\] as the dimensions of matrix $\varSigma _{n}$ go to infinity at the same pace. Such quantities arise in the study of functionals of $\varSigma _{n}\varSigma _{n}^{*}$ which do not only depend on the eigenvalues of $\varSigma _{n}\varSigma _{n}^{*}$, 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.
LA - eng
KW - random matrix; empirical distribution of the eigenvalues; Stieltjes transform; bilinear forms; spectral norm; wireless communication system; signal processing
UR - http://eudml.org/doc/272084
ER -

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