Spectral Properties Of Large-dimensional Random Matrix Involved In Canonical Correlation Analysis

This paper is mainly concerned with the spectral properties of large-dimensional random matrices involved in canonical correlation analysis,including the limit spectral distribution and the description of its support set,the limit and asymptotic distribution of spiked eigenvalues,and the central limit theorem for linear spectral statistics.First of all,some research backgrounds and research methods in random matrix theory are introduced in the first chapter of this paper.The section focused on research background can be divided into: the research history and current situation of random matrices based on four random matrix models and their corresponding spectral properties;The history of correlation analysis and its connection with random matrices;To facilitate readers’ comprehension,we conclude this section by showing the structure framework and collecting some notational conventions.In the second section,we enumerate the research methods of three common concepts:limit spectral distribution Support sets,spiked eigenvalues,and linear spectral statistics.Then in Chapter 2,we give the results and proofs of limit spectral distribution of noncentral Fisher matrix and the necessary and sufficient conditions for determining the support set,which is the theoretical basis for studying the noncentral Fisher matrix.Under Gaussian assumption,according to the functional relationship between the sample canonical correlation coefficient and the eigenvalues of the noncentral Fisher matrix,Chapter 3 is aimed as giving the limit and asymptotic distribution of the spiked eigenvalues of the noncentral Fisher matrix under the assumption of a normal population,in order to obtain the asymptotic result of the sample canonical correlation coefficient.As a side-product,we get an intermediate result—the limit and asymptotic normality of the spiked eigenvalues of the noncentral sample covariance matrix.In some specific problems,this result has its own importance,but we will not pursue the extension of this aspect in this paper.In particular,our assumption about the spectral structure of the noncentral parameter matrix here is quite weak,so it has to be sacrificed at the expense of the general distribution assumption.We complement central limit theorem for linear spectral statistics of generalized Beta matrix and its proof in Chapter 4.The application and simulation of the full text are postponed to Chapter 5,as are the numerical tests for conjecture results.To relax the limitation of Gaussian assumption in Chapter 3,we explored a feasible research idea and proved the universality of the limit spectral distribution of the sample canonical correlation coefficient matrix,and then elaborated on the problems that still need to be solved in the future.