Some Results Of Limiting Spectral Distribution Of Large Dimensional Random Matrices

In applied mathematics, probability and statistics, and modern physical area, the spectral theory of large dimensional random matrix is a prevalent research topic. It has wide spread application s in many subject areas. In this thesis, we mainly concentrate on the limiting spectral distribution of large dimensional random matrices.In chapter I, we will introduce the significance of the subject, the research back-ground, the research methods and the innovation points related to this thesis.Chapter 2 includes three aspects, and it is devoted to the limiting spectral distribution of three classes of large dimensional random matrices. We will generalize the results of the limiting spectral distribution of Wigner matrix (?) and the sample covariance matrix Sn=1/NYnYn* which have been studied by Bai[7] and the result of the limiting spectral distribution of the information-plus-noise type sample covariance matrix Cn =1/N(Rn + ?Yn)(Rn+ ?Yn)* which has been studied by Xie[27]. In their work, besides some regular assumptions, they assumed an additional assumption that Exij2= 1(i < j)· In this paper, we will weaken this assumption to Exij2 = ?ij2(i<j), but 1/n?i=1n|1/n?j=1n?ij2-1|?0, and extend the results of them to a more general case.In chapter 3, we will consider the model Bn = 1/NXnXn*Tn, where Tn is Hermitian non-negative definite, and as i n??, the empirical spectral distribution of FTN?H,a.s.,where H is a non-random probability distribution function on [0,?). Let Xn be a com-plex random matrix independent of Tn. Silverstein[24] studied the the limiting spectral distribution of Bn under the conditions that all the entries of Xn are i.i.d. random vari-ables. In this thesis, we will expand the conclusion under the conditions that the entries of Xn are i.i.d. to the case that the columns of Xn are i.i.d. random vectors. Then we will prove that, almost surely, the empirical spectral distribution of Bn converges weakly to a non-random distribution whose Stieltjes transform satisfies a certain equation.In chapter 4, we will present a short summary and list some potential research plans in our next investigations.