Robust modifications of U-statistics and applications to covariance estimation problems

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 exponential 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.