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

Philippe Loubaton (Université Paris-Est Marne-la-Vallée)
Pascal Vallet (Université Paris-Est Marne-la-Vallée)

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.

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Pages: 1934-1959

Publication Date: October 20, 2011

DOI: 10.1214/EJP.v16-943

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