This paper considers Tyler's M-estimator of the covariance matrix in high dimensional elliptical distributions. We focus on the setting in which both the sample size n and the dimension p are finite. We show that as long as n is larger than pln(p), the squared Frobenius norm of the error decays like pln(p)/n with high probability. In particular, this means that Tyler's estimator in elliptical distribution behaves like the traditional sample covariance in Gaussian distributions. This contribution extends recent similar results in the asymptotic regime where n is infinite, as well as the double asymptotic regime where both n and p are infinite.
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