Large Dimensional Random Matrix Spectral Distribution Of The Limit Theory And Its Application

Large dimensional random matrix is a popular research topic in the probability and statistics. Recently, as the computer science develops and the information blasts, we need to deal with large data sets with high dimensions in many fields. Thus, large dimensional random matrices have been paid more attention to than before. Applications of large dimensional random matrix exist in many research fields, such as theoretical physics, wireless communication, finance and genetic statistics. The development of large dimensional random matrix has been in close contact with these subjects. In this thesis we focused on the research of spectral distribution of large dimensional matrix and its application in multivariate statistical analysis.The background knowledge and research status of random matrix are briefly introduced in chapter 1. Some classic matrices are introduced, such as Wigner matrix, sample covariance matrix. One of the key problem in random matrix theory is to investigate the convergence of empirical spectral distribution whose definition iswhere λ₁, …,λ_n are the eigenvalues of random matrix A. The limiting distribution of F^A(x) is usually non-random. For the research of the spectral distribution ,there are two very important aspects that we will discuss: one is the explicit form of limiting spectral distrib-ution(chapter 2); the other is the convergence rates of spectral distributions(chapter 3 and chapter 4).Further, we will apply the results of chapter 2 in the factor model(chapter 5).In chapter 2,we discussed the following problem: let S_n =1/nX_nX_n^* be the sample covariance matrices and T_n be a sequence of Hcrmitian matrices independent of X_n.Does the Limiting Spectral Distribution (LSD) of the product of a sample covariance matrix with an Hermitian matrix S_nT_n exist? Our answer is positive. Especially, when the Hcrmitian matrix is a Wigner matrix W_n, the explicit expression of the density function of LSD of S_nW_n is derived.Following, we discuss convergence rates of spectral distribution. In chapter 3, we improve the convergence rates of spectral distributions that large-dimensional sample covariance matrices of size p x n tends to the Marcenko - Pastur distribution, under the assumption that the entries have a finite eighth moment. Especially, we showed that the expected spectral distributions of large-dimensional sample covariance matrices of size p x n tends to the limiting distribution with the dimension sample size ratio y = y_n = p/n at the rate of O(n-1/6)...