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October 2011 Asymptotic properties of eigenmatrices of a large sample covariance matrix
Z. D. Bai, H. X. Liu, W. K. Wong
Ann. Appl. Probab. 21(5): 1994-2015 (October 2011). DOI: 10.1214/10-AAP748


Let Sn = 1/n XnXn* where Xn = {Xij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_{n}(\mathbf{t}_{1},\mathbf{t}_{2},\sigma)=\sqrt{p}({\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}(S_{n}+\sigma I)^{-1}{\mathbf{x}}_{n}(\mathbf{t}_{2})-{\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}{\mathbf{x}}_{n}(\mathbf{t}_{2})m_{n}(\sigma))$ 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 ${\mathbb{C}}^{p}$, having indices t1 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 Sn is asymptotically close to that of a Haar-distributed unitary matrix.


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Z. D. Bai. H. X. Liu. W. K. Wong. "Asymptotic properties of eigenmatrices of a large sample covariance matrix." Ann. Appl. Probab. 21 (5) 1994 - 2015, October 2011.


Published: October 2011
First available in Project Euclid: 25 October 2011

zbMATH: 1234.15013
MathSciNet: MR2884057
Digital Object Identifier: 10.1214/10-AAP748

Primary: 15A52
Secondary: 15A18‎ , 60F05

Keywords: central limit theorems , Haar distribution , Linear spectral statistics , Marčenko–Pastur law , Random matrix , Sample covariance matrix , semicircular law

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.21 • No. 5 • October 2011
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