Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices (Q2196219)

Let \(\mathbf{Y}=\mathbf{\Gamma X}\) be the data matrix, where \(\mathbf{X}\) be a \((p+l)\times n\) random matrix whose entries are independent with mean means and unit variances and \(\mathbf{\Gamma}\) is a \(p\times(p+l)\) deterministic matrix under condition \(l/p\rightarrow0\). Let \(\mathbf{\Sigma}=\mathbf{\Gamma}\mathbf{\Gamma}^\intercal\) be the population covariance matrix. The sample covariance matrix in such a case is \[ S_n=\frac{1}{n}\mathbf{Y}\mathbf{Y}^\intercal=\frac{1}{n}\mathbf{\Gamma X}\mathbf{X}^\intercal\mathbf{\Gamma}^\intercal. \] Let \(\mathbf{V}\mathbf{\Lambda}^{1/2}\mathbf{U}\) denote the singular value decomposition of matrix \(\mathbf{\Gamma}\), where \(\mathbf{V}\) and \(\mathbf{U}\) are orthogonal matrices and \(\mathbf{\Lambda}\) is a diagonal matrix consisting in descending order eigenvalues \(\mu_1\geqslant\mu_2\geqslant\ldots\geqslant\mu_p\) of matrix \(\mathbf{\Sigma}\). Authors of the paper suppose that there are \(K\) spiked eigenvalues that are separated from the rest. They assume that eigenvalues \(\mu_1\geqslant\ldots\geqslant\mu_K\) tends to infinity, while the other eigenvalues \( \mu_{K+1}\geqslant\ldots\geqslant\mu_p\) are bounded. In the paper, the asymptotic behaviour is considered of the spiked eigenvalues and the largest non-spiked eigenvalue. The limiting normal distribution for the spiked sample eigenvalues is established. The limiting \textit{Tracy-Widom} law for the largest non-spiked eigenvalues is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are considered.