Introduction to a large random matrix theory

This article provides an introduction to large random matrix theory, aimed at a nonspecialist audience. We state and prove Marcenko-Pastur's theorem which describes the asymptotic 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 Marcenko-Pastur theorem. We also present covariance matrices with general population covariance matrices and spiked models. We finally give an application of the theory to wireless communication.