Random matrix theory and its application in high-dimensional statistics

Abstract

This thesis mainly focuses on several classical random matrices under some special settings, which has wide applications in modern science. We study the limiting spectral distribution of the m by m upper-left corner of an n by n Haar-invariant unitary matrix, which converges to the circular law as m goes to infinity with m over n goes to 0 or converges to the arc law as m over n goes to 1. Secondly we investigate the random eigenvalues coming from the beta-Laguerre ensemble with parameter p, which is a generalization of the Wishart matrices of parameter (n,p). In the case when the sample size n is much smaller than the dimension p, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble which is a generalization of the Wigner matrices. As corollaries, we get that the largest and smallest eigenvalues of the complex Wishart matrix are asymptotically independent; the limiting distribution of the condition numbers; a test procedure for the spherical hypothesis test. In addition, we prove the large deviation principles for three basic statistics: the largest eigenvalue, the smallest eigenvalue and the empirical distribution of eigenvalues, and we also demonstrate that the limiting spectral distribution converges to the semicircle law as a corollary. Finally, we use a modified statistic to test the covariance structure for the n by p random matrix where p is much larger than n. Both the law of the large number and the limiting distribution of the statistic are derived. Under certain conditions, we also obtain the rate of convergence for the asymptotic distribution. Furthermore, we also use the same type of statistics to test the banded covariance structure, and show under some conditions, it holds with the similar properties.