Finite-dimensional Mixed Symmetric Ensembles Random Matrix Theory And Its Applications

In this thesis,random matrix theory such a powerful mathematical tool is used to study the finite-dimensional mixed symmetric ensembles.In this thesis,we derive new conclusions of finite-dimensional mixed symmetric ensembles and apply them to various scenarios.This thesis carries out the following research,which is mainly divided into three parts:The first part is the establishment of the unified form of the joint eigenvalues distribution of the mixed symmetric ensembles.We first propose a unified expression of the joint probability density function of ordered eigenvalues for the mixed symmetric ensembles,and then derive joint distributions of eigenvalues for four types of crossed ensembles in detail: Laguerre unitary and orthogonal cross ensembles,Gaussian orthogonal and unitary cross ensembles,Gaussian symplectic and unitary cross ensembles,Circle orthogonal and unitary cross ensembles.In addition,the joint probability density functions of eigenvalues for special cases such as Gaussian Unitary ensemble and Wishart matrix are derived,which shows that the unified expression proposed in this thesis can be used for some classical symmetric ensembles after particularity.The second part is the derivation and application of marginal probability density function（PDF）.Based on the unified form of joint eigenvalues distribution,we propose a method to derive the marginal distribution of eigenvalues,and derive the closed expression of the marginal PDF of any single ordered eigenvalue for the first time.In the derivation,we also propose a formula for calculating multiple integrals and give the proof.In addition,we provide a unified expression on the marginal PDF of any subset of ordered eigenvalues.The expression of marginal PDF of any single ordered eigenvalue is applied to the performance analysis of multiple-input multiple-output（MIMO）eigen-mode communication system,and solves the problem of the exact expression of the Symbol Error Rate（SER）of each subchannel under Nakagami- fading.The marginal PDFs of the largest and the smallest eigenvalues are applied to the characterization of gap probability.The joint marginal PDF of any subset of ordered eigenvalues is applied to the level spacing distribution.All results are verified by Monte Carlo simulations.The third part is the derivation and application of marginal cumulative distribution function（CDF）.Based on the unified expression of the joint eigenvalues distribution,we propose a method to derive the exact expression of the marginal CDF of any single ordered eigenvalue.And in the derivation,we prove a multiple integral calculation equation based on the uniqueness of probability distribution.In addition,a unified asymptotic expansion method is proposed and used to derive the asymptotic expansion expression of the marginal CDF for LUE-LOE cross ensemble.The exact expression and asymptotic expression are applied to the performance analysis of MIMO system.We derive the exact expression of diversity gain and array gain of the system under Hoyt fading for the first time.All results proposed are verified by simulations.