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 for N vector observations y₁, ..., y_N ∈ R^m 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 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.