Large information plus noise random matrix models and consistent subspace estimation in large sensor networks

In array processing, a common problem is to estimate the angles of arrival of $K$ deterministic sources impinging on an array of $M$ antennas, from $N$ observations of the source signal, corrupted by gaussian noise. The problem reduces to estimate a quadratic form (called "localization function") of a certain projection matrix related to the source signal empirical covariance matrix. Recently, a new subspace estimation method (called "G-MUSIC") has been proposed, in the context where the number of available samples $N$ is of the same order of magnitude than the number of sensors $M$. In this context, the traditional subspace methods tend to fail because the empirical covariance matrix of the observations is a poor estimate of the source signal covariance matrix. The G-MUSIC method is based on a new consistent estimator of the localization function in the regime where $M$ and $N$ tend to $+\infty$ at the same rate. However, the consistency of the angles estimator was not adressed. The purpose of this paper is to prove the consistency of the angles of arrival estimator in the previous asymptotic regime. To prove this result, we show the property that the singular values of M x N Gaussian information plus noise matrix escape from certain intervals is an event of probability decreasing at rate O(1/N^p) for all p. A regularization trick is also introduced, which allows to confine these singular values into certain intervals and to use standard tools as Poincar\'e inequality to characterize any moments of the estimator. These results are believed to be of independent interest

Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights

We prove in a direct fashion that a multidimensional probability measure is determinate if the higher dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in the associated L_p spaces for all finite p. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under the measure. We give practical examples of such weights, based on their classification. As in the one dimensional case, the results on determinacy of measures supported on R^n lead to sufficient conditions for determinacy of measures supported in a positive convex cone, i.e. the higher dimensional analogue of determinacy in the sense of Stieltjes.Comment: 20 pages, LaTeX 2e, no figures. Second and final version, with minor corrections and an additional section on Stieltjes determinacy in arbitrary dimension. Accepted by The Annals of Probabilit