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    Asymptotic properties of eigenmatrices of a large sample covariance matrix

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    Let Sn=1nXnXnS_n=\frac{1}{n}X_nX_n^* where Xn={Xij}X_n=\{X_{ij}\} is a p×np\times n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1,t2,σ)=p(xn(t1)(Sn+σI)1xn(t2)xn(t1)xn(t2)mn(σ))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\sigma>0 and mn(σ)=dFyn(x)x+σm_n(\sigma)=\int\frac{dF_{y_n}(x)}{x+\sigma} where Fyn(x)F_{y_n}(x) is the Mar\v{c}enko--Pastur law with parameter yn=p/ny_n=p/n; which converges to a positive constant as nn\to\infty, and xn(t1){\mathbf {x}}_n(\mathbf {t}_1) and xn(t2){\mathbf {x}}_n(\mathbf {t}_2) are unit vectors in Cp{\Bbb{C}}^p, having indices t1\mathbf {t}_1 and t2\mathbf {t}_2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1,t2,σ)Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma) converges weakly to a (2m+1)(2m+1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of SnS_n is asymptotically close to that of a Haar-distributed unitary matrix.Comment: Published in at http://dx.doi.org/10.1214/10-AAP748 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org
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