On asymptotics of eigenvectors of large sample covariance matrix

Let {X ij }, i, j = ..., be a double array of i.i.d. complex random variables with EX 11 = 0, E|X 11 | 2 = 1 and E|X 11 | 4 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the square root of a nonnegative definite matrix T n and X n is the n x N matrix of the upper-left comer of the double array. The matrix An can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T n , or as a multivariate F matrix if T n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of An, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if { X ij } and T n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of An are proved to have Gaussian limits, which suggests that the eigenvector matrix of An is nearly Haar distributed when T n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

[1]  J. W. Silverstein.  Weak Convergence of random functions defined by the eigenvectors of sample covariance matrices , 1990 .

[2]  P. Krishnaiah,et al.  A Limit Theorem for the Eigenvalues of Product of Two Random Matrices , 2003 .

[3]  Z. Bai,et al.  On the limit of the largest eigenvalue of the large dimensional sample covariance matrix , 1988 .

[4]  J. W. Silverstein,et al.  No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices , 1998 .

[5]  Z. Bai,et al.  CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data , 2017, Statistical Papers.

[6]  J. W. Silverstein.  On the eigenvectors of large dimensional sample covariance matrices , 1989 .

[7]  Z. Bai,et al.  Limit of the smallest eigenvalue of a large dimensional sample covariance matrix , 1993 .

[8]  Jack W. Silverstein.  Some limit theorems on the eigenvectors of large dimensional sample covariance matrices , 1984 .

[9]  K. Wachter.  The Limiting Empirical Measure of Multiple Discriminant Ratios , 1980 .

[10]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[11]  Shlomo Shamai,et al.  Spectral Efficiency of CDMA with Random Spreading , 1999, IEEE Trans. Inf. Theory.

[12]  Dag Jonsson.  Some limit theorems for the eigenvalues of a sample covariance matrix , 1982 .

[13]  Zhidong Bai,et al.  CONVERGENCE RATE OF EXPECTED SPECTRAL DISTRIBUTIONS OF LARGE RANDOM MATRICES PART II: SAMPLE COVARIANCE MATRICES , 2008 .

[14]  Z. Bai,et al.  Convergence Rate of Expected Spectral Distributions of Large Random Matrices. Part I. Wigner Matrices , 1993 .

[15]  Yonina C. Eldar,et al.  On the asymptotic performance of the decorrelator , 2003, IEEE Trans. Inf. Theory.

[16]  David Tse,et al.  Linear Multiuser Receivers: Effective Interference, Effective Bandwidth and User Capacity , 1999, IEEE Trans. Inf. Theory.

[17]  J. W. Silverstein.  Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices , 1995 .

[18]  Z. Bai.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW , 1999 .

[19]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[20]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[21]  K. Wachter.  The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements , 1978 .

[22]  Jian-Feng Yao,et al.  Convergence Rates of Spectral Distributions of Large Sample Covariance Matrices , 2003, SIAM J. Matrix Anal. Appl..

[23]  J. W. Silverstein,et al.  EXACT SEPARATION OF EIGENVALUES OF LARGE DIMENSIONAL SAMPLE COVARIANCE MATRICES , 1999 .