On the isolated eigenvalues of large Gram random matrices with a fixed rank deformation
Résumé
Consider the matrix $\Sigma_n = n^{-1/2} X_n D_n^{1/2} + P_n$ where the matrix $X_n \in \C^{N\times n}$ has Gaussian standard independent elements, $D_n$ is a deterministic diagonal nonnegative matrix, and $P_n$ is a deterministic matrix with fixed rank. Under some known conditions, the spectral measures of $\Sigma_n \Sigma_n^*$ and $n^{-1} X_n D_n X_n^*$ both converge towards a compactly supported probability measure $\mu$ as $N$ and $n$ converge to infinity at the same rate. In this paper, we prove that finitely many eigenvalues of $\Sigma_n\Sigma_n^*$ may stay outside the support of $\mu$ in the large dimensional regime. The existence and locations of these eigenvalues in any compact interval outside the support of $\mu$ are studied. The fluctuations of the extreme isolated eigenvalues of $\Sigma_n\Sigma_n^*$ are also analyzed. The results find applications in the fields of signal processing and radio communications.
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