CONVERGENCE RATES OF EIGENVECTOR EMPIRICAL SPECTRAL DISTRIBUTION OF LARGE DIMENSIONAL SAMPLE COVARIANCE MATRIX

• 成果类型: Article
• 署名作者: Xia, Ningning; Qin, Yingli; Bai, Zhidong
• 署名单位: Northeast Normal University - China; Northeast Normal University - China; National University of Singapore; University of Waterloo
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1154
• 发表日期: 2013
• 页码: 2572-2607
• 关键词: Principal component analysis; LARGEST EIGENVALUE; asymptotics; limit; LAW

摘要

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of sample covariance matrix and the Marcenko-Pastur distribution function is of order O(N-1/2). Given that data dimension n to sample size N ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed eta > 0, the convergence rates of VESD are O(N-1/4) in probability and O(N-1/4+eta) almost surely, requiring finite 8th moment of the underlying distribution.