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

Research paper by **Philippe Loubaton, Pascal Vallet**

Indexed on: **29 Sep '11**Published on: **29 Sep '11**Published in: **Mathematics - Probability**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

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