Some Properties Of Complex Random Variables And Complex Random Quadratic Forms

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.