On the consistency of likelihood penalization methods in large sensor networks

This paper is devoted to the problem of source detection with large sensor networks, in a context where the number of available samples N and the number of antennas M are of the same order of magnitude. We focus here on the popular likelihood penalization (LP) methods, such as Minimum Description Length (MDL) or Akaike Information Criterion (AIC). Such methods have been widely studied in the context where N >;>; M, and in particular the consistency of the MDL and the inconsistency of the AIC estimator were established in the asymptotic regime where N → ∞ while M remains constant. We propose here an analysis in the asymptotic regime where M;N both converge to ∞ at the same rate, and using results from random matrix theory, we establish conditions on the penalty term to ensure consistency of LP methods in this latter regime. As a consequence, we deduce that the MDL method is always inconsistent while the AIC method can be consistent in certain situations.