Convergence Of The Empirical Spectral Distribution Function Of Large Dimensional Random Matrices

In this paper, we focus on the convergence of the empirical spectral distribu-tion function of large dimensional random matrices, including the limiting spectral distribution of large dimensional random matrices, the convergence rates of the empirical spectral distribution function of large dimensional random matrices and the central limit theorem of linear spectral statistics of large dimensional random matrices.In the first section, we will introduce some definitions and backgrounds of random matrices, especially we will introduce the most important two types of random matrices:Wigner matrices and covariance matrices. And also we will give this thesis’s outline in this section. In the second section, we will intro-duce the main results and tools about proving the limiting of empirical spec-tral distribution function, the convergence rates of the empirical spectral distri-bution function and the central limit theorems of linear spectral statistics. What is more, in this part, we will introduce some basic results such as the limiting spectral distribution of Wigner matrices, sample covariance matrices, F-matrices, and so on. The convergence rates of the empirical spectral distribution function of Wigner matrices and M-P type matrices will be given in Section3and Sec-tion4respectively. Especially, the results about the convergence rates of the empirical spectral distribution function of Wigner matrices have been published at<Electronic Journal of Probability> and the results about the convergence rates of the empirical spectral distribution function of M-P type matrices have been published at<Stochastic Processes and their Applications>. A consistent k-ernel estimator of the limiting spectral distribution of M-P type matrices will be introduced in Section5and also the central limit theorem of the kernel estimator is proved, these results have been submitted to<Transactions of the AMS>. The last section focus on the limiting empirical spectral distribution function and the central limit theorem of linear spectral statistics of a widely used type of random matrices in multivariate statistical analysis which are called Beta matrices, and these results have been submitted to<Bernoulli>.