  We study the problem of outlier robust high-dimensional mean estimation under
a finite covariance assumption, and more broadly under finite low-degree moment
assumptions. We consider a standard stability condition from the recent robust
statistics literature and prove that, except with exponentially small failure
probability, there exists a large fraction of the inliers satisfying this
condition. As a corollary, it follows that a number of recently developed
algorithms for robust mean estimation, including iterative filtering and
non-convex gradient descent, give optimal error estimators with
(near-)subgaussian rates. Previous analyses of these algorithms gave
significantly suboptimal rates. As a corollary of our approach, we obtain the
first computationally efficient algorithm with subgaussian rate for
outlier-robust mean estimation in the strong contamination model under a finite
covariance assumption.
