The outliers among the singular values of large rectangular random matrices with additive fixed rank deformation

Abstract

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 $N,n\to\infty$ with $N/n\to c>0$. In this paper, it is proved that finitely many eigenvalues of $\Sigma_n\Sigma_n^*$ may stay away from the support of $\mu$ in the large dimensional regime. The existence and locations of these outliers in any connected component of $\R - \support(\mu)$ are studied. The fluctuations of the largest outliers of $\Sigma_n\Sigma_n^*$ are also analyzed. The results find applications in the fields of signal processing and radio communications.