On Probability Limit Theorems For Spectral Statistics Of Random Matrices

The spectral theory of random matrix is a popular research topic in both ap-plied mathematics and probability and statistics, and its applications exist in many research fields. In this thesis, we mainly concentrate on the probability limit prop-erties of the spectral statistics of random matrices. It mainly contains the following three parts:that is, the precise asymptotics of the spectral statistics of random matrices, limiting spectral distribution of sample covariance matrices, and central limit theorem for the linear spectral statistics of large random matrices.Some of the background knowledge and research status, including research methods are briefly introduced in Chapter1.In Chapter2. we will establish the precise asymptotics of the spectral statistics to describe the limiting spectral properties of random matrices, which are inspired by the theory of the complete convergence for independent sum of random variables. In the first part, we will give the precise asymptotics of the spectral statistics in the context of complete convergence and conditional moment, which generated by the empirical spectral measure of the Wigner matrices and the sample covariance matrices respectively. And the second part will focus on the precise asymptotics of the largest eigenvalues of HβE and LβE in the context of conditional moment.In Chapter3. we will discuss the limiting spectral distribution of the information-plus noise type sample covariance matrices Cn=1/N(Rn+σXn)(Rn+σXn)*. which has been studied by Dozier&Silverstein(2007a) under the assumption that the en-tries of Xn are i.i.d. random variables. In this paper, we will weaken the conditions on the the entries of Xn. and consider two different cases. The first case is the entries of Xn are independent but non-identically distributed random variables and satisfy a Lindeberg condition. Another case is the columns of Xn are i.i.d. ran-dom vectors. Under the assumptions of the two cases, we will prove that, almost surely, the empirical spectral distribution of Cn converges weakly to a non-random distribution whose Stieltjes transform satisfies a certain equation. Chapter4is devoted to the central limit theorem(CLT) of the linear spectral statistics. It includes three aspects, the first one is CLT of linear spectral statistics of large dilute Wigner matrices when the Fourier transform of the test function sat-isfies a regular smooth condition. The second one is to consider the convergence of the spectral empirical process instructed by the linear spectral statistics of complex Wigner matrices. There is a remarkable work due to Bai&Yao(2005), who proved that the empirical spectral process indexed by the test function, converges weakly in finite dimensions to a Gaussian process. In their work, besides some regular assump-tions, they assumed an additional assumption that Exij2=0(i<j). This paper will remove this additional condition and extends the result of Bai&Yao(2005) to a more general case. The last one is to investigate the limiting properties of the eigenvalue counting function for the eigenvalues up to the edge of the spectrum of the sample covariance matrices. By using the Four Moment Theorem and Rigidity Theorem of eigenvalues, we will extend the CLT of the eigenvalue counting function of LUE obtained by Su(2006) and moderate deviation principle of the determinan-tal point process established by Doring&Eichelsbacher(2011) to the more general sample covariance matrices.At last, we will present a short summary and discussion of open problems to be considered in further investigations.