  Let $Y$ be a $d$-dimensional random vector with unknown mean $\mu$ and
covariance matrix $\Sigma$. This paper is motivated by the problem of designing
an estimator of $\Sigma$ that admits tight deviation bounds in the operator
norm under minimal assumptions on the underlying distribution, such as
existence of only 4th moments of the coordinates of $Y$. To address this
problem, we propose robust modifications of the operator-valued U-statistics,
obtain non-asymptotic guarantees for their performance, and demonstrate the
implications of these results to the covariance estimation problem under
various structural assumptions.
