Spectral Properties Of Large Dimensional Random Sample Covariance Matrices

This thesis is concerned on two kinds of sample covariance matrices.One is quater-nion sample covariance matrices.Our aim is to study the convergence of empirical spec-tral distributions of large dimensional quaternion sample covariance matrices and the ex-treme eigenvalues of large dimensional quaternion sample covariance matrices.The other is general sample covariance matrices.We focus on the central limit theorem of linear spectral statistics of large dimensional general sample covariance matrices.At the beginning of the first chapter,we introduce the backgrounds and some defini-tions of random matrices.Next we show the definitions and properties of quaternions.In the sequel,we present two common method:the Stieltjes transform and the characteristic function.We give the outline of the thesis at the end of this chapter.Chapter 2 and Chapter 3 establish two properties of large dimensional quaternion sample covariance matrices.In Chapter 2,our goal is from the convergence of the em-pirical spectral distributions of large dimensional quaternion sample covariance matrices to the convergence of their Stieltjes transforms by the method of Stieltjes transform.We show that their limiting spectral distribution is M-P law.Chapter 3 adopts the method of graphic theory and shows that the extreme eigenvalues of large dimensional quaternion sample covariance matrices almost surely converge to the two end points of M-P law.Chapter 4 is about general sample covariance matrices.The first half only considers the normal distribution since the normal random variables keep unchanged through or-thogonal transformation.The last half extends the result from the Gaussian case to the general case through comparing their characteristic functions.