The Annals of Probability

On asymptotics of eigenvectors of large sample covariance matrix

Z. D. Bai, B. Q. Miao, and G. M. Pan

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Abstract

Let {Xij}, i, j=…, be a double array of i.i.d. complex random variables with EX11=0,E|X11|2=1 and E|X11|4<∞, and let $A_{n}=\frac{1}{N}T_{n}^{{1}/{2}}X_{n}X_{n}^{*}T_{n}^{{1}/{2}}$, where Tn1/2 is the square root of a nonnegative definite matrix Tn and Xn is the n×N matrix of the upper-left corner 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 Tn, or as a multivariate F matrix if Tn 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 {Xij} and Tn 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 Tn is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

Article information

Source
Ann. Probab., Volume 35, Number 4 (2007), 1532-1572.

Dates
First available in Project Euclid: 8 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1181334252

Digital Object Identifier
doi:10.1214/009117906000001079

Mathematical Reviews number (MathSciNet)
MR2330979

Zentralblatt MATH identifier
1162.15012

Subjects
Primary: 15A52 60F15: Strong theorems 62E20: Asymptotic distribution theory
Secondary: 60F17: Functional limit theorems; invariance principles 62H99: None of the above, but in this section

Keywords
Asymptotic distribution central limit theorems CDMA eigenvectors and eigenvalues empirical spectral distribution function Haar distribution MIMO random matrix theory sample covariance matrix SIR Stieltjes transform strong convergence

Citation

Bai, Z. D.; Miao, B. Q.; Pan, G. M. On asymptotics of eigenvectors of large sample covariance matrix. Ann. Probab. 35 (2007), no. 4, 1532--1572. doi:10.1214/009117906000001079. https://projecteuclid.org/euclid.aop/1181334252


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