Please use this identifier to cite or link to this item: https://doi.org/10.1214/10-AAP748
Title: Asymptotic properties of eigenmatrices of a large sample covariance matrix
Authors: Bai, Z.D.
Liu, H.X. 
Wong, W.K.
Keywords: Central limit theorems
Haar distribution
Linear spectral statistics
Marčenko-Pastur law
Random matrix
Sample covariance matrix
Semicircular law
Issue Date: Oct-2011
Citation: Bai, Z.D., Liu, H.X., Wong, W.K. (2011-10). Asymptotic properties of eigenmatrices of a large sample covariance matrix. Annals of Applied Probability 21 (5) : 1994-2015. ScholarBank@NUS Repository. https://doi.org/10.1214/10-AAP748
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.
Source Title: Annals of Applied Probability
URI: http://scholarbank.nus.edu.sg/handle/10635/125049
ISSN: 10505164
DOI: 10.1214/10-AAP748
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