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

Journal Article

### Source:

Markov Processes and Related Fields, Volume 20 (2014)

### Abstract:

Consider the matrix $Σn = n^{−1/2}X_nD_n^{1/2} + P_n$ where the matrix $X_n ∈ C^{N×n}$ 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 $ΣnΣ∗ n$ and $n^{−1}X_nD_nX^∗_n$ both converge towards a compactly supported probability measure $µ$ as $N, n → \infty$ with $N/n → c > 0$. In this paper, it is proved that finitely many eigenvalues of $ΣnΣ∗n$ may stay away from the support of µ in the large dimensional regime. The existence and locations of these outliers in any connected component of $R − supp(µ)$ are studied. The fluctuations of the largest outliers of $ΣnΣ∗ n$ are also analyzed. The results find applications in the fields of signal processing and radio communications.

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