ABSTRACT
This chapter explains the main notions, results and details of classical as well as techniques and deals with large random matrices. It summarizes some important results regarding the characterization of the support of the eigenvalues of a sample covariance matrix and the position of the individual eigenvalues of a sample covariance matrix. The chapter provides the estimators of functionals of the eigenvalues of a population covariance matrix based on the observation of a sample covariance matrix. It investigates large dimensional sample covariance matrix models with population covariance matrix composed of a few eigenvalues with large multiplicities. Random matrix theory deals with the study of matrix-valued random variables. Several approaches can be used to derive the Marcenko-Pastur law. The chapter describes the original technique proposed by Marcenko and Pastur is based on a fundamental tool, the Stieltjes transform. It shows that, under mild assumptions on the random matrix model, all eigenvalues are asymptotically contained within the limiting support.
