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Asymptotic properties of eigenmatrices of a large sample covariance matrix. (English) Zbl 1234.15013

Authors’ abstract: Let \(S_n = 1/n\, X_nX_n^*\) where \(X_n = \{X_{ij}\}\) is a \(p \times n\) 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 \(\sigma > 0\) and \(m_n(\sigma) = \int d F_{y_n}(x)/(x + \sigma)\) where \(F_{y_n}(x)\) is the Marčenko-Pastur law with parameter \(y_n = p/n\); which converges to a positive constant as \(n \to \infty\), and \(\mathbf x_n(\mathbf t_1)\) and \(\mathbf x_n(\mathbf t_2)\) are unit vectors in \({\mathbb{C}}^p\), having indices \(\mathbf t_1\) and \(\mathbf t_2\), ranging in a compact subset of a finite-dimensional Euclidean space. We prove that the sequence \(Y_n(\mathbf t_1, \mathbf t_2, \sigma)\) 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.

MSC:

15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
15A18 Eigenvalues, singular values, and eigenvectors
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