## 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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