This article studies the limiting behavior of a class of robust population
covariance matrix estimators, originally due to Maronna in 1976, in the regime
where both the number of available samples and the population size grow large.
Using tools from random matrix theory, we prove that, for sample vectors made
of independent entries having some moment conditions, the difference between
the sample covariance matrix and (a scaled version of) such robust estimator
tends to zero in spectral norm, almost surely. This result can be applied to
various statistical methods arising from random matrix theory that can be made
robust without altering their first order behavior.Comment: to appear in IEEE Transactions on Information Theor