PROPERTIES OF EIGENVALUES AND EIGENVECTORS OF LARGE-DIMENSIONAL SAMPLE CORRELATION MATRICES

• 成果类型: Article
• 署名作者: Yin, Yanqing; Ma, Yanyuan
• 署名单位: Chongqing University; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1802
• 发表日期: 2022
• 页码: 4763-4802
• 关键词: linear spectral statistics; central-limit-theorem; Covariance matrices; CONVERGENCE; interference; asymptotics; edge; clt

摘要

This paper is to study the properties of eigenvalues and eigenvectors of high-dimensional sample correlation matrices. We first improve the result of Jiang (Sankhya over bar 66 (2004) 35-48), Xiao and Zhou (J. Theoret. Probab. 23 (2010) 1-20) and the Theorem 1 of El Karoui (Ann. Appl. Probab. 19 (2009) 2362-2405), both concerning the limiting spectral distribution and the ex-treme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matri-ces. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influ-enced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrices is not asymptotically Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigen-vectors of large sample covariance matrices. This theorem improves the main results in Bai, Miao and Pan (Ann. Probab. 35 (2007) 1532-1572), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (Ann. Appl. Probab. 18 (2008) 1232-1270), which relaxed the Gaus-sian like fourth moment requirement but assumes the maximum entries of the projection vector converge to 0 uniformly. We illustrate the usefulness of the theoretical results through an application in communications.