Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence

We consider estimating the shared mean of a sequence of heavy-tailed random variables taking values in a Banach space. In particular, we revisit and extend a simple truncation-based mean estimator first proposed by Catoni and Giulini. While existing truncation-based approaches require a bound on the raw (non-central) second moment of observations, our results hold under a bound on either the central or non-central pth moment for some p ∈ (1,2]. Our analysis thus handles distributions with infinite variance. The main contributions of the paper follow from exploiting connections between truncation-based mean estimation and the concentration of martingales in smooth Banach spaces. We prove two types of time-uniform bounds on the distance between the estimator and unknown mean: line-crossing inequalities, which can be optimized for a fixed sample size n, and iterated logarithm inequalities, which match the tightness of line-crossing inequalities at all points in time up to a doubly logarithmic factor in n. Our results do not depend on the dimension of the Banach space, hold under martingale dependence, and all constants in the inequalities are known and small.