The Annals of Applied Probability

Asymptotic properties of eigenmatrices of a large sample covariance matrix

Z. D. Bai, H. X. Liu, and W. K. Wong

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 5 (2011), 1994-2015.

Dates
First available in Project Euclid: 25 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1319576615

Digital Object Identifier
doi:10.1214/10-AAP748

Mathematical Reviews number (MathSciNet)
MR2884057

Zentralblatt MATH identifier
1234.15013

Subjects
Primary: 15A52
Secondary: 60F05: Central limit and other weak theorems 15A18: Eigenvalues, singular values, and eigenvectors

Keywords
Random matrix central limit theorems linear spectral statistics sample covariance matrix Haar distribution Marčenko–Pastur law semicircular law

Citation

Bai, Z. D.; Liu, H. X.; Wong, W. K. Asymptotic properties of eigenmatrices of a large sample covariance matrix. Ann. Appl. Probab. 21 (2011), no. 5, 1994--2015. doi:10.1214/10-AAP748. https://projecteuclid.org/euclid.aoap/1319576615


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