Large deviations of condition numbers and extremal eigenvalues of random matrices

Abstract

Random matrix theory found applications in many areas, for instance in statistics random matricesareusedtoanalysemultivariatedataandtheireigenvaluesareusedinhypothesistesting. Spectral properties of random matrices have been studied extensively in the literature dealing with both the bulk case (involving all the eigenvalues) and the extremal case (addressing the maximal and minimal eigenvalues). In this thesis two types of sequences of random matrices areconsidered: thefirsttypeisthesequenceofsamplecovariancematricesandthesecondtype isthesequenceof β´Laguerre(orWishart)ensembles,forwhichlargedeviationsoftheirextremal cases are studied. These two types of sequences of random matrices contain the classical Wishart matrices. The thesis can be divided into two parts. The first part is on the study of large deviations of condition numbers defined as ratios of maximal and the minimal eigenvalues. This is done based on suitable analysis and estimates of the joint density function of all eigenvalues. The secondpartdealswithlargedeviationsofindividualmaximalandminimaleigenvalue,andthe approach consists of suitable eigenvalue concentration inequalities and Laplace’s method. Itisremarkedthatforthosetwotypesofsequencesofrandommatricesconsideredinthisthesis, two scenarios are investigated: either one of the dimension size and the sample size is much larger than the other one, or the two sizes are comparable