Almost Sure Localization of the Eigenvalues in a Gaussian Information Plus Noise Model. Application to the Spiked Models.

Abstract

Let $S$ be a $M$ times $N$ random matrix defined by $S = B + \sigma W$ where $B$ is a uniformly bounded deterministic matrix and where $W$ is an independent identically distributed complex Gaussian matrix with zero mean and variance $1/N$ entries. The purpose of this paper is to study the almost sure location of the eigenvalues of the Gram matrix $SS^*$ when $M$ and $N$ converge to infinity such that the ratio $M/N$ converges towards a constant $c > 0$. The results are used in order to derive, using an alternative approach, known results concerning the behavior of the largest eigenvalues of $SS^*$ when the rank of $B$ remains fixed and $M$ and $N$ converge to infinity.