Almost sure localization of the eigenvalues in a Gaussian information plus noise model – Application to the spiked models

Abstract

Let ΣN be a M × N random matrix defined by ΣN = BN + σWN where BN is a uniformly bounded deterministic matrix and where WN 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 λ1,N ≥ … ≥ λM,N of the Gram matrix ΣNΣ*N when M and N converge to +∞ such that the ratio cN = M/N converges towards a constant c > 0. The results are used in order to derive, using an alternative approach, known results concerning the behaviour of the largest eigenvalues of ΣNΣ*N when the rank of BN remains fixed and M,N tend to +∞. © 2011 Applied Probability Trust.