  Tyler's and Maronna's M-estimators, as well as their regularized variants,
are popular robust methods to estimate the scatter or covariance matrix of a
multivariate distribution. In this work, we study the non-asymptotic behavior
of these estimators, for data sampled from a distribution that satisfies one of
the following properties: 1) independent sub-Gaussian entries, up to a linear
transformation; 2) log-concave distributions; 3) distributions satisfying a
convex concentration property. Our main contribution is the derivation of tight
non-asymptotic concentration bounds of these M-estimators around a suitably
scaled version of the data sample covariance matrix. Prior to our work,
non-asymptotic bounds were derived only for Elliptical and Gaussian
distributions. Our proof uses a variety of tools from non asymptotic random
matrix theory and high dimensional geometry. Finally, we illustrate the utility
of our results on two examples of practical interest: sparse covariance and
sparse precision matrix estimation.
