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On asymptotics of eigenvectors of large sample covariance matrix. (English) Zbl 1162.15012

Let \(X_n=(X_{ij})\) be an \(n\times N\) matrix of independent, identically distributed complex random variables and \(T_n\) be an \(n\times n\) nonnegative definite Hermitian matrix with a square root \(T_n^{1/2}\), and \(A_n=\displaystyle\frac{1}{N}T_n^{1/2}X_nX_n^*T_n^{1/2}\), in which both the dimension \(n\) and the sample size \(N\) are large. The authors show the central limit theorem for those analytic functions over the support of the limiting spectral distribution of \(A_n\) under the condition that \[ EX_{11}=0,\quad E|X_{11}|^2=1, \quad E|X_{11}|^4<\infty. \] Moreover, if \(\{X_{ij}\}\) and \(T_n\) are either real or complex and some additional moment assumptions are made then the linear spectral statistic defined by the eigenvectors of \(A_n\) are proved to have Gaussian limits, which suggests that the eigenvector matrix of \(A_n\) is nearly Haar distributed when \(T_n\) is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

MSC:

15B52 Random matrices (algebraic aspects)
62E20 Asymptotic distribution theory in statistics
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
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References:

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