大维随机样本协方差阵的谱性质

发布时间：2018-07-05 14:16

本文选题：四元数样本协方差矩阵 + 极限谱分布函数　；参考：《东北师范大学》2016年博士论文

【摘要】：本篇论文我们主要研究了两类样本协方差矩阵,一类是四元数样本协方差矩阵,研究内容为经验谱分布函数的收敛和矩阵最大最小特征值的极限;另一类是广义样本协方差矩阵,研究内容为线性谱统计量的中心极限定理。首先在第一章中介绍了随机矩阵的背景及基本概念,四元数的基本定义和性质,还有要用到的方法,章节末尾给出了论文构架。接着在第二章和第三章给出了四元数样本协方差阵的两个谱性质。第二章用Stieltjes变换方法,将经验谱分布函数的收敛转变为研究其Stieltjes变换的收敛,并得出四元数样本协方差矩阵的极限谱分布函数为M-P律。第三章用图论的方法证明了四元数样本协方差矩阵的极值特征值分别几乎处处收敛到M-P律的两端点。第四章研究对象转变为广义样本协方差矩阵。前半部分先考虑正态情况,主要利用了正态随机变量经过正交变换后分布不变的性质,后半部分通过比较正态和非正态情形下的特征函数来完成证明。
[Abstract]:In this paper, we mainly study two kinds of sample covariance matrices, one is the quaternion sample covariance matrix, the other is the convergence of empirical spectrum distribution function and the limit of the maximum and minimum eigenvalues of the matrix. The other is the generalized sample covariance matrix, which is the central limit theorem of linear spectral statistics. In the first chapter, the background and basic concept of random matrix, the basic definition and properties of quaternion, and the methods to be used are introduced. At the end of the chapter, the framework of the paper is given. Then, in the second and third chapters, we give two spectral properties of quaternion sample covariance matrix. In the second chapter, by using Stieltjes transformation method, the convergence of empirical spectral distribution function is transformed into the convergence of Stieltjes transformation, and the limit spectral distribution function of quaternion sample covariance matrix is obtained as M-P law. In chapter 3, we prove that the extremum eigenvalues of quaternion sample covariance matrix converge almost everywhere to the two ends of M-P law by the method of graph theory. In chapter 4, the object is transformed into the generalized sample covariance matrix. In the first half of the paper, the normal condition is considered first, the distribution of the normal random variables is invariant after orthogonal transformation, and the second half is proved by comparing the eigenfunctions in normal and non-normal cases.
【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：C829.2

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