On independence and large deviations for sublinear expectations

TL;DR

This paper corrects a previous large deviation principle in the framework of sublinear expectations, showing the original result only holds under stronger independence assumptions and providing a revised version.

Contribution

It identifies errors in prior work on large deviations under sublinear expectations and offers a corrected theorem with clarified conditions.

Findings

The original large deviation principle is incorrect in general.

The rate function cannot always be derived from the Fenchel transform.

A corrected version of the principle is established under stronger independence assumptions.

Abstract

We prove by counterexample that a large deviation principle established by Chen and Feng [{\em Comm. Statist. Theory Methods} {\bf 45} (2016), 400--412] in the framework of sublinear expectations is incorrect. That implies that the rate function cannot, in general, be obtained by computing the Fenchel transform of the cumulant generating function, as is the case for ordinary probabilities. We derive a corrected version of that result and show that the original presentation holds under a stronger independence assumption.