The random matrix regime of Maronna’s estimator for observations corrupted by elliptical noise

Abstract

We study the behavior of Maronna’s robust scatter estimator C ˆ N ∈ C N × N built from a sequence of observations y 1 , … , y n lying in a K -dimensional signal subspace of the N -dimensional complex field corrupted by heavy tailed noise, i.e., y i = A N s i + x i , where A N ∈ C N × K and x i is drawn from an elliptical distribution. In particular, we prove under mild assumptions that the robust scatter matrix can be characterized by a random matrix S ˆ N that follows a standard random model as the population dimension N , the number of observations n , and the rank of A N grow to infinity at the same rate. Our results are of potential interest for statistical theory and signal processing.