發(fā)布時間：2018-07-05 14:16

  本文選題：四元數(shù)樣本協(xié)方差矩陣 + 極限譜分布函數(shù)��； 參考：《東北師范大學》2016年博士論文

【摘要】：本篇論文我們主要研究了兩類樣本協(xié)方差矩陣,一類是四元數(shù)樣本協(xié)方差矩陣,研究內(nèi)容為經(jīng)驗譜分布函數(shù)的收斂和矩陣最大最小特征值的極限;另一類是廣義樣本協(xié)方差矩陣,研究內(nèi)容為線性譜統(tǒng)計量的中心極限定理。首先在第一章中介紹了隨機矩陣的背景及基本概念,四元數(shù)的基本定義和性質(zhì),還有要用到的方法,章節(jié)末尾給出了論文構(gòu)架。接著在第二章和第三章給出了四元數(shù)樣本協(xié)方差陣的兩個譜性質(zhì)。第二章用Stieltjes變換方法,將經(jīng)驗譜分布函數(shù)的收斂轉(zhuǎn)變?yōu)檠芯科銼tieltjes變換的收斂,并得出四元數(shù)樣本協(xié)方差矩陣的極限譜分布函數(shù)為M-P律。第三章用圖論的方法證明了四元數(shù)樣本協(xié)方差矩陣的極值特征值分別幾乎處處收斂到M-P律的兩端點。第四章研究對象轉(zhuǎn)變?yōu)閺V義樣本協(xié)方差矩陣。前半部分先考慮正態(tài)情況,主要利用了正態(tài)隨機變量經(jīng)過正交變換后分布不變的性質(zhì),后半部分通過比較正態(tài)和非正態(tài)情形下的特征函數(shù)來完成證明。
[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.
【學位授予單位】：東北師范大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：C829.2

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