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

Philippe Loubaton (1) , Pascal Vallet (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

Let Sigma(N) be a M x N random matrix defined by Sigma(N) = B(N) + sigma W(N) where B(N) is a uniformly bounded deterministic matrix and where W(N) 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 (lambda) over cap (1,N) >= ... >= (lambda) over cap (M,N) of the Gram matrix Sigma(N)Sigma(N)* when M and N converge to +infinity such that the ratio c(N) = 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 Sigma(N)Sigma(N)* when the rank of B(N) remains fixed and M, N tend to +infinity.

Domaines: Physique mathématique [math-ph]

Philippe Loubaton, Pascal Vallet. Almost sure localization of the eigenvalues in a Gaussian information plus noise model - Application to the spiked models. Electronic Journal of Probability, 2011, 16 (?), pp.1934--1959. ⟨hal-00692258⟩