Topics On Spectral Analysis Of Covariance Matrices And Its Application

The spectral theory of random matrix is a popular research topic in both ap-plied mathematics and probability and statistics,and it serves as a very power-ful tool to deal with statistical applications particularly for high dimensional time series problems. In this thesis,we mainly concentrate on probability limit prop-erties of spectral distribution of renormalized separable sample covariance matri-ces of spatial-temporal processes as well as that of renormalized sample and auto-covariance matrices of linear processes.Basic ideas and knowledge that related to the topics in this thesis are introduced in Chapter1.In Chapter2,Consider the data matrix Yn=Ap1/2.XnBn1/2,where Xn has i.i.d mean0,variance1and finite fourth moment. The sample covariance ma-trix of the model are given as Sn=Ap1/2XnBnXn*Ap1/2relates very closely to the separable covariance model. In fact,the covariance structure of vectorized data matriX Yn=Ap1/2XnBn1/2can be expressed in the form of Ap(?)Bn,where (?) denotes Kronecker product. In that context,the rows of Yn correspond to spa-tial locations while the columns represent the observation times. Under the set-ting that p/nâ†'0,∥Sn-ESn∥â†'a.s.0.From B ai and Yin[8]study,when Ap=Jp,Bn=In as well as p,nâ†'∞such that p/nâ†'0时，the renormalized matrices√n/p(Sn-ESn)=√n/p(n-1XnXn*-Ip)haS the sampe limiting spec-tral distribution as that of a p×p Wigner matrix Wp.Then,for a general sample covariance matrices Sn=n-1Ap1/2XnXn*Ap1/2,where.Xn i.i.d has zero mean,unit variance and finite fourth moments,Pan and Gao[53]and Bao[18]derive the LSD of√n/p(n-1Ap1/2XnXn*Ap1/2-Ap).We actually consider a even more general model of the form when p,nâ†'∞and p/nâ†'0,the ESD of Cn converges almost Surely to a nonrandom distribution F,such that the Stieltjes transform of F satisfies s(z)= and We also derive the density of the F and all these results are utilized to propose a test for the co-variance structure of the data where the null hypothesis is that the joint covariance matrix is of the form Ap(?) Bnfor⑧denoting the Kronecker product。The perfor-mance of this test is illustrated through a simulation study. Moreover,we show that when Xn has sub-Gaussian entries and Ap is a mixture of finite point mass, the empirical distribution of{√n/p(λj(Sn)-n-1tr(Bn)-(Ap))}jP=l converges to F(x)=∑jm=1cjfsc(x;√B2cjαj)。 In Chapter3,we investigates the limiting spectral behavior of renormalized sample covariance and auto-covariance matrices of a p—dimension linear time series of the form xt=Zt+∑l=1∞AlZt-l,t∈Z where Zt,t∈Z is a sequence of independent p-dimension real or complex random vector with zero mean and unit variance and fnite fourth moments.Deefine the renormalized covariance matrices where andwhen both the dimension and sample size are very large but the dimension in-creasing much slower compared with the sample size.We show that the empirical spectral distribution of its renormalized sample covariance and auto-covariance ma-trices converges a.s.to a nonrandom limit distribution FT supported on R+and its Stieltjes transform satisfies where where βT(z, a) is the unique solution satisfying that βT(z, a) is Stieltjes kernel. The proof requires the assumptions that pxp Hermitian matrices{Al}lq=1are simultane-ously diagonalizable and satisfying that Σl2∥Al∥<∞. This provides an innovative perspective on investigating the behavior of high dimensional linear time series and provides a way on model diagnosis and prediction as well as the coefficients estima-tion.At last,we present a short summary and potential research plans considered in further investigations.