Limiting Spectral Behaviors For Several Classes Of High Dimensional Random Matrix Models And Their Statistical Applications

With the rapid development of computing science,the processing and analysis of high-dimensional massive data is available,and the research on the theory and application of high dimensional data is in the ascendant.The research on high dimensional random matrix spectral behaviors provides statistical theory support for high dimensional data processing and analysis,which brings about a more specific and accurate results on the data processing and analysis."Random matrix" is a kind of matrix whose elements are random variables in some probability spaces.In addition,the main purpose of spectral analysis of random matrices is to study the theoretical properties and distribution laws for the eigenvalues and eigenvectors of random matrices.Starting from the properties of the spectrum for high dimensional random matrices,the main results are as follows:Firstly,the limiting spectral distribution(LSD)of auto-cross covariance matrix(ACVM)generated by high dimensional vector autoregressive moving average model(VARMA)is studied by Stieltjes transformation.In most cases,there exists no explicit forms for the LSD.To deal with this problem,the thesis presents a kind of kernel nonparametric estimation for the limiting distribution and density functions.Besides,the consistency and convergence rate are proved.Because the limiting density(distribution)function always contains the information of the regression coefficients,an approach named "Median of Means" is proposed to estimate such coefficients,which is proved to be consistent and robust.Secondly,in view of the high dimensional VARMA model which is time dependent,the limiting distribution of the extremum eigenvalue for the ACVM is proved to be "Tracy-Widom(TW)type".The processing is based on the Large Deviation Theorem and the method of Green Matching.Moreover,the corresponding results of non-Gaussian population are given.Specifically,what kinds of moment conditions should be met so that the limiting distribution of the extremum eigenvalue is also given by "TW distribution".In addition,compared with the traditional trace test,a variety of cases are examined in the simulation study,such as the order test,parameter test etc.Finally,by incorporating the effect of spiked eigenvalues,the thesis proposes a supplement of the CLT for LSSs under the high dimensional spiked covariance model(named by H_pCLT),which eliminates the systematic deviation between the empirical distributionand the asymptotic distribution on their mean and variance parameters.Besides,the elimination effect of this deviation are shown by comparative tests.