The random matrix regime of Maronna's M-estimator for observations corrupted by elliptical noises

This article studies the behavior of the Maronna robust scatter estimator $\hat{C}N\in \mathbb{C}^{N\times N}$ of a sequence of observations $y1,...,yn$ which is composed of a $K$ dimensional signal drown in a heavy tailed noise, i.e $yi=AN si+xi$ where $AN \in \mathbb{C}^{N\times K}$ and $xi$ is drawn from elliptical distribution. In particular, we prove that as the population dimension $N$, the number of observations $n$ and the rank of $AN$ grow to infinity at the same pace and under some mild assumptions, the robust scatter matrix can be characterized by a random matrix $\hat{S}_N$ that follows a standard random model. Our analysis can be very useful for many applications of the fields of statistical inference and signal processing.