  We present an estimator of the covariance matrix $\Sigma$ of random
$d$-dimensional vector from an i.i.d. sample of size $n$. Our sole assumption
is that this vector satisfies a bounded $L^p-L^2$ moment assumption over its
one-dimensional marginals, for some $p\geq 4$. Given this, we show that
$\Sigma$ can be estimated from the sample with the same high-probability error
rates that the sample covariance matrix achieves in the case of Gaussian data.
This holds even though we allow for very general distributions that may not
have moments of order $>p$. Moreover, our estimator can be made to be optimally
robust to adversarial contamination. This result improves the recent
contributions by Mendelson and Zhivotovskiy and Catoni and Giulini, and matches
parallel work by Abdalla and Zhivotovskiy (the exact relationship with this
last work is described in the paper).
