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 matrices are used to analyse multivariate data and their eigenvalues are used in hypothesis testing. 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 are considered: the first type is the sequence of sample covariance matrices, and the second type is the sequence of β-Laguerre (or Wishart) ensembles, for which large deviations of their extremal 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 second part deals with large deviations of individual maximal and minimal eigenvalue, and the approach consists of suitable eigenvalue concentration inequalities and Laplace’s method.It is remarked that for those two types of sequences of random matrices considered in this thesis, 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.