We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a nonasymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.
"Sub-Gaussian mean estimators." Ann. Statist. 44 (6) 2695 - 2725, December 2016. https://doi.org/10.1214/16-AOS1440