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

Abstract

Let $\boldsymbolΣ_N$ be a $M \times N$ random matrix defined by $\boldsymbolΣ_N = \mathbf{B}_N + σ\mathbf{W}_N$ 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 $\hatλ_{1,N} \geq ... \geq \hatλ_{M,N}$ of the Gram matrix ${\boldsymbol Σ}_N {\boldsymbol Σ}_N^*$ 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 ${\boldsymbol Σ}_N {\boldsymbol Σ}_N^*$ when the rank of $\mathbf{B}_N$ remains fixed when $M$ and $N$ converge to $+\infty$.