Exact separation of eigenvalues of large dimensional sample covariance matrices

• 成果类型: Article
• 署名作者: Bai, ZD; Silverstein, JW
• 署名单位: National University of Singapore; North Carolina State University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1999
• 页码: 1536-1555
• 关键词: limiting spectral distribution

摘要

Let B-n = (1/N)T-n(1/2) X-n (XnTn1/2)-T-* where X-n is n x N with i.i.d. complex standardized entries having finite fourth moment, and T-n(1/2) is a Hermitian square root of the nonnegative definite Hermitian matrix T-n. It was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of T-n, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large rt. For these n the interval corresponds to one that separates the eigenvalues of T-n. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of B-n and T-n lying on one side of their respective intervals are identical for all large n.