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

In this paper, we propose an alternative proof for the uniqueness of Maronna’s M -estimator of scatter [1] for N vector observations y1, . . . ,yN ∈ R under a mild constraint of linear independence of any subset of m of these vectors. This entails in particular almost sure uniqueness for random vectors yi 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.