On the Convergence of Maronna's M-Estimators of Scatter

In this letter, we propose an alternative proof for the uniqueness of Maronna's M-estimator of scatter [1] for vector observations y(1),...,y(N) is an element of R-m under a mild constraint of linear independence of any subset of of these vectors. This entails in particular almost sure uniqueness for random vectors y(i) with a density as long as N > m. This approach allows to establish further relations that demonstrate that a properly normalized Tyler's M-estimator of scatter [2] can be considered as a limit of Maronna's M-estimator. More precisely, the contribution is to show that each M-estimator, verifying some mild conditions, converges towards a particular Tyler's M-estimator. These results find important implications in recent works on the large dimensional (random matrix) regime of robust M-estimation.