An introduction to large random matrix theory

An introduction to large random matrix theory

Jamal Najim

Université Paris-Est Marne La Vallée et CNRS Laboratoire d’Informatique Gaspart Monge Champs sur Marne, France

Corresponding Author Email: 
jamal.najim@univ-mlv.fr
Page: 
161-222
|
DOI: 
https://doi.org/10.3166/TS.33.161-222
Received: 
21 April 2015
| |
Accepted: 
17 November 2015
| | Citation

OPEN ACCESS

Abstract: 

This article provides an introduction to large random matrix theory, aimed at a non- specialist audience. We state and prove Marcˇenko-Pastur’s theorem which describes the asymp- totic spectrum of a large covariance matrix. We introduce the Stieltjes transform and associated techniques; we also introduce specific techniques for matrices with gaussian entries, which in particular provide a short proof for the isotropic Marcˇenko-Pastur theorem. We also present co- variance matrices with general population covariance matrices and spiked models. We finally give an application of the theory to wireless communication.

Keywords: 

random matrix theory.

Extended abstract
1. Introduction
2. Transformée de Stieltjes
3. Identités matricielles et inégalités
4. Théorème de Marcˇenko-Pastur
5. Le modèle de grandes matrices de covariance
6. Petites perturbations
7. Grandes matrices aléatoires et communications numériques
Remerciements
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