ASYMPTOTIC PROPERTIES OF EIGENMATRICES OF A LARGE SAMPLE COVARIANCE MATRIX

• 成果类型: Article
• 署名作者: Bai, Z. D.; Liu, H. X.; Wong, W. K.
• 署名单位: Northeast Normal University - China; Northeast Normal University - China; National University of Singapore; Hong Kong Baptist University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP748
• 发表日期: 2011
• 页码: 1994-2015
• 关键词: limiting spectral distribution; eigenvectors

摘要

Let S(n) = 1/nX(n)X*(n) where X(n) = {X(ij)} is a p x n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Y(n)(t(1), t(2), sigma) = root p(x(n)(t(1))*(S(n) + sigma I)(-1)x(n)(t(2)) - x(n)(t(1))*x(n)(t(2)) m(n)(sigma)) in which sigma > 0 and m(n)(sigma) = integral dF(yn)(x)/x+sigma where F(yn)(x) is the Marcenko-Pastur law with parameter y(n) = p/n; which converges to a positive constant as n -> infinity, and x(n)(t(1)) and x(n)(t(2)) are unit vectors in C(p), having indices t(1) and t(2), ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Y(n)(t(1), t(2), sigma) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S(n) is asymptotically close to that of a Haar-distributed unitary matrix.