【作者】 王英；

【导师】 熊玮；

【作者基本信息】 湖南大学 ， 概率论与数理统计， 2020， 硕士

【摘要】 本文主要是在已有复随机分布的基础上,结合实随机分布的相关性质,利用矩阵变换,极坐标变换得到了几个多维复随机分布的结论,提出了复随机变量模的最值分布定理;用矩阵迹与期望的可交换性给出了Hermite复随机二次型期望的一个简单表示;根据Hermite矩阵可酉对角化的性质推广了Hermite复随机二次型的表示;细化了Hermite正态复随机二次型分布定理的证明,得到了Hermite正态复随机二次型的矩母函数与特征函数,并给出了相关证明.本文从以下几个章节对复随机分布进行研究.第一章,主要介绍了实随机与复随机分布的发展史;概述了部分复随机分布的研究现状与本文的主要研究内容;给出了文中用到的一些记号.第二章,主要概括了复随机变量及复随机向量的一些性质,并给出了部分性质的详细证明.第三章,首先应用两个相互独立的复随机变量的性质,推广了复正态随机变量分布的可加性;其次,结合矩阵变换的性质得到了复正态随机向量子向量的分布;最后,利用极坐标变换法得到了复随机变量模的最值分布定理,并推广了复随机变量的Cauchy-Schwarz不等式与三角不等式.第四章,首先结合矩阵迹与期望的可交换性,得到了Hermitie复随机二次型期望的一个简单表达式;其次,用Hermite矩阵可酉对角化的性质及前辈对实随机二次型表示的研究方法,分别给出了Hermite复随机二次型在奇异与非奇异情况下的表示;最后,研究了Hermite正态复随机二次型Q=Z~HRZ与Hermite正态复随机二次表达式Q=Z~HRZ+a~TMZ+b的分布,并得到了它们的矩母函数与特征函数.更多还原

【Abstract】 In this thesis,several results about multi-dimensional complex random distributions are obtained.These results are based on the complex random distributions,in combination with the related properties of real random distributions.We proved these results by using matrix transformations and polar transformations.The main results are as follows.The first is about the maximum distribution of the modules of complex random variables.The second result gives a simple expression for the expectation of an Hermitian complex random quadratic form,by using the commutativity of matrix trace and expectation.The third result is about a generalization of the expression of Hermite complex random quadratic forms,by using the fact that Hermitian matrices can be unitarily diagonalized.Finally,we studied the distributions of Hermitian normal complex random quadratic forms,and obtained the moment generating functions and characteristic functions of such complex random quadratic forms.The contents of each chapter are given as follows.In chapter 1,we gave a short history of the development of real and complex random distributions and a brief discription of the current research status of some complex random distributions,summariezed the main results of this thesis,and introduced some preliminaries and notations.In chapter 2,we summarized some properties of complex random variables and vectors,and gave detailed proofs of several properties.In chapter 3,several results about complex random variables are proved.First we obtained the additivity of complex random normal distributions,by using some properties about two independent complex random variables.Then we studied the distributions of subvectors of complex normal vectors,by using matrix transformations.Finally,we obtained the distribution of the maximum of the modules of complex random variables,by using the method of polar coordinates transformation;and we showed the Cauchy-Schwarz inequality and the triangle inequality for complex random variables.In chapter 4,we studied the complex random quadratic forms.First,we obtained a simple expression for the expectation of an Hermitian complex random quadratic form,by using the commutativity of matrix trace and expectation.Then we gave the representations of normal Hermitian complex random quadratic forms in the singular and non-singular cases,by using the fact that Hermitian matrices can be unitarily diagonalized,in combinintion with the methods used in previous studies on the representations of real random quadratic forms.Finally,we studied the distributions of Hermitian normal complex random quadratic forms Q=Z~HRZ and quadratic expression Q=Z~HRZ+a~TMZ+b,and we obtained the moment generating functions and the characteristic functions of such forms.更多还原