Estimation of the covariance structure of heavy-tailed distributions
We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distribution, such as existence of moments of only low order. While estimation of covariance matrices corresponding to sub-Gaussian distributions is well-understood, much less in known in the case of heavy-tailed data. As K. Balasubramanian and M. Yuan write 1 , "data from real-world experiments oftentimes tend to be corrupted with outliers and/or exhibit heavy tails. In such cases, it is not clear that those covariance matrix estimators .. remain optimal" and "..what are the other possible strategies to deal with heavy tailed distributions warrant further studies." We make a step towards answering this question and prove tight deviation inequalities for the proposed estimator that depend only on the parameters controlling the "intrinsic dimension" associated to the covariance matrix (as opposed to the dimension of the ambient space); in particular, our results are applicable in the case of highdimensional observations.
The paper provides a truncation estimator for the covariance matrix. The idea is simple yet useful, enjoying very good theoretical properties. I strongly welcome this contribution to the robust statistics literature.
I have only several very minor comments/suggestions.
(1) The authors should discuss relevance to covariance estimation under elliptical distributions, like ECA proposed by Han and Liu (JASA, 2017+) (which is also a robust estimator of the covariance when the eigenvalues are bounded; see, e.g., https://arxiv.org/pdf/1705.06427.pdf). See below for more details.  
(2) Can we derive estimation accuracy under other norms (such as L_infty norm and restricted spectral norm, defined in the above ECA paper)? These alternative norms are crucial in high dimensional statistics (e.g., graphical model estimation and sparse PCA). See the ECA paper for an example.
(3) Centeredness is not crucial in covariance matrix estimation as long as the data are iid: We could use the pairwise difference X_i-X_j, which is centered no matter whether X_i is centered or not. This alternative should also be more efficient than the proposed one since it avoids estimating the mean. See again the ECA paper for discussions. Relative comments have also been put in Chen, Gao, and Ren (AoS, 2017+), which the authors cited.