TY - JOUR

T1 - PROPERTIES OF EIGENVALUES AND EIGENVECTORS OF LARGE-DIMENSIONAL SAMPLE CORRELATION MATRICES

AU - Yin, Yanqing

AU - Ma, Yanyuan

N1 - Funding Information:
Funding. Yanqing Yin’s work is partially supported by a project under Grant NSFC11801234. Yanyuan Ma’s work is partially supported by grants from the National Institute of Health.
Publisher Copyright:
© 2022 Institute of Mathematical Statistics. All rights reserved.

PY - 2022/12

Y1 - 2022/12

N2 - 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 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 extreme 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 matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced 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 eigenvectors 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 Gaussian 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.

AB - 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 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 extreme 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 matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced 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 eigenvectors 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 Gaussian 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.

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U2 - 10.1214/22-AAP1802

DO - 10.1214/22-AAP1802

M3 - Article

AN - SCOPUS:85147744389

SN - 1050-5164

VL - 32

SP - 4763

EP - 4802

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 6

ER -