Robust high-dimensional Gaussian and bootstrap approximations for trimmed sample means

Most of the modern literature on robust mean estimation focuses on designing estimators which obtain optimal sub-Gaussian concentration bounds under minimal moment assumptions and sometimes also assuming contamination. This work looks at robustness in terms of Gaussian and bootstrap approximations, mainly in the regime where the dimension is exponential on the sample size. We show that trimmed sample means attain - under mild moment assumptions and contamination - Gaussian and bootstrap approximation bounds similar to those attained by the empirical mean under light tails. We apply our results to study the Gaussian approximation of VC-subgraph families and also to the problem of vector mean estimation under general norms, improving the bounds currently available in the literature.