On asymptotics of eigenvectors of large sample covariance matrix

• 成果类型: Article
• 署名作者: Bai, Z. D.; Miao, B. Q.; Pan, G. M.
• 署名单位: Northeast Normal University - China; Northeast Normal University - China; Chinese Academy of Sciences; University of Science & Technology of China, CAS; National University of Singapore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000001079
• 发表日期: 2007
• 页码: 1532-1572
• 关键词: dimensional random matrices; SPECTRAL DISTRIBUTIONS; Empirical distribution; LIMIT-THEOREMS; eigenvalues; CONVERGENCE

摘要

Let {X-ij}, i, j = .... be a double array of i.i.d. complex random variables with EX11 =0, E vertical bar X-11 vertical bar(2) = 1 and E vertical bar X-11 vertical bar(4) < infinity, and let A(n) = 1/N T-n(1/2) X-n (XnTn1/2)-T-*, where T-n(1/2) is the square root of a nonnegative definite matrix T-n and X-n is the n x N matrix of the upper-left corner of the double array. The matrix A(n) can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T-n, or as a multivariate F matrix if T-n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of A(n), a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if {X-ij} and T-n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of A(n) are proved to have Gaussian limits, which suggests that the eigenvector matrix of A(n) is nearly Haar distributed when T-n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.