#### Document Type

Journal Article

#### Department/Unit

Department of Economics

#### Title

Asymptotic properties of eigenmatrices of a large sample covariance matrix

#### Language

English

#### Abstract

Let Sn = 1/n XnXn where Xn = {X ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t 2,σ)=√p(xn(t1) *(Sn +σI)-1xn(t2)-x n(t1)*xn(t2)m n(σ)) in which σ > 0 and mn(σ)= ∫dFyn(x)/x+σ where Fyn(x) is the Marčenko-Pastur law with parameter yn = p/n; which converges to a positive constant as n → ∞ and xn(t1) and xn(t2) are unit vectors in ℂp, having indices t 1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S n is asymptotically close to that of a Haar-distributed unitary matrix. © 2011 Institute of Mathematical Statistics.

#### Keywords

Central limit theorems, Haar distribution, Linear spectral statistics, Marčenko-Pastur law, Random matrix, Sample covariance matrix, Semicircular law

#### Publication Date

2011

#### Source Publication Title

Annals of Applied Probability

#### Volume

21

#### Issue

5

#### Start Page

1994

#### End Page

2015

#### Publisher

Institute of Mathematical Statistics

#### DOI

10.1214/10-AAP748

#### Link to Publisher's Edition

http://dx.doi.org/10.1214/10-AAP748

#### ISSN (print)

10505164

#### Recommended Citation

Bai, Z. D.,
H. X. Liu,
and
W. K. Wong.
"Asymptotic properties of eigenmatrices of a large sample covariance matrix."
*Annals of Applied Probability*
21.5
(2011): 1994-2015.