Almost sure localization of the eigenvalues in a gaussian information plus noise model. Applications to the spiked models

Abstract

Let ${\mathbf{\Sigma }}_N$ be a $M\times N$ random matrix defined by ${\mathbf{\Sigma }}_N={\mathbf{B}}_N+\sigma {\mathbf{W}}_N$ where ${\mathbf{B}}_N$ is a uniformly bounded deterministic matrix and where ${\mathbf{W}}_N$ is an independent identically distributed complex Gaussian matrix with zero mean and variance $\frac{1}{N}$ entries. The purpose of this paper is to study the almost sure location of the eigenvalues ${\hat{\lambda }}_{1,N}\ge ...\ge {\hat{\lambda }}_{M,N}$ of the Gram matrix ${\mathbf{\Sigma }}_N{\mathbf{\Sigma }}_N^{\ast }$ when $M$ and $N$ converge to $+\infty$ such that the ratio $c_N=\frac{M}{N}$ converges towards a constant $c>0$. The results are used in order to derive, using an alernative approach, known results concerning the behaviour of the largest eigenvalues of ${\mathbf{\Sigma }}_N{\mathbf{\Sigma }}_N^{\ast }$ when the rank of ${\mathbf{B}}_N$ remains fixed when $M$ and $N$ converge to $+\infty$.

19 pages, 1 figure, Accepted for publication in Electronic Journal of Probability

References in corpus (1)

• Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices

Cited by in corpus (2)

• The Random Matrix Regime of Maronna's M-estimator with elliptically distributed samples
• Robust spiked random matrices and a robust G-MUSIC estimator