Exponential Concentration Inequalities For Independent Random Vectors Under Sublinear Expectations

TL;DR

This paper develops exponential concentration inequalities for the sample mean of independent random vectors under sublinear expectations, extending previous variance bounds to sub-Gaussian and sub-exponential regimes with optimal rates.

Contribution

It introduces the first exponential concentration inequalities for multivariate sample means under sublinear expectations, including dimension-free bounds and optimal sub-Gaussian rates.

Findings

Proves a general concentration principle reducing vector tail bounds to scalar martingale inequalities.

Establishes a sub-Gaussian tail bound for the distance from the sample mean to the expectation set.

Provides a Bernstein inequality interpolating between sub-Gaussian and sub-exponential regimes.

Abstract

Li and Hu recently established variance-type O(1/n) bounds for the sample mean of independent random vectors under sublinear expectations. We extend their results to the exponential concentration regime. For bounded, independent R^d-valued random vectors under a regular sublinear expectation, we prove: (i) a general concentration principle that reduces vector-valued tail bounds to scalar martingale inequalities via a three-layer architecture; (ii) an Azuma-Hoeffding inequality showing that the distance from the sample mean to the Minkowski average of the expectation sets has sub-Gaussian tails; (iii) a Bernstein inequality incorporating the variance parameter of Li and Hu, interpolating between sub-Gaussian and sub-exponential regimes; (iv) a dimension-free bound replacing the exponential covering prefactor with a polynomial one via the matrix Freedman inequality; and (v) an explicit construction demonstrating that the sub-Gaussian rate is optimal. To the best of our knowledge, these constitute the first exponential concentration inequalities for the multivariate sample mean under sublinear expectations in terms of the set-valued distance to the Minkowski average.