Berry-Esseen Bounds and Moderate Deviations for Catoni-Type Robust Estimation

Abstract: A powerful robust mean estimator introduced by Catoni (2012) allows for mean estimation of heavy-tailed data while achieving the performance characteristics of classical mean estimator for sub-Gaussian data. While Catoni's framework has been widely extended across statistics, stochastic algorithms, and machine learning, fundamental asymptotic questions regarding the Central Limit Theorem and rare event deviations remain largely unaddressed. In this paper, we investigate Catoni-type robust estimators in two contexts: (i) mean estimation for heavy-tailed data, and (ii) linear regression with heavy-tailed innovations. For the first model, we establish the Berry--Esseen bound and moderate deviation principles, addressing both known and unknown variance settings. For the second model, we demonstrate that the associated estimator is consistent and satisfies a multi-dimensional Berry-Esseen bound.