On independence and large deviations for sublinear expectations

TL;DR

This work scrutinizes large deviation principles within sublinear expectation frameworks, specifically for negatively dependent sequences. It first provides a counterexample to Chen and Feng’s claimed LDP lower bound, demonstrating that the rate function cannot in general be obtained via the Fenchel--Legendre transform of the cumulant generating function. Building on a sublinear Gärtner–Ellis approach, the authors establish a corrected LDP for negatively dependent variables, proving an upper bound that always holds and a lower bound that is valid under independence (where the cumulant conjugates coincide). The results reveal a more intricate link between limit theorems under sublinear expectations and the classical probabilistic setting, clarifying when Fenchel-transform-based rate functions are appropriate. Overall, the paper delineates the limits of extending classical LDPs to sublinear frameworks and provides a corrected, rigorous foundation for future work in capacity-based large deviations.

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.

Figures (1)

• Figure 1: The function $I_p$ is plotted in red, and the function $I$ is plotted in blue.

Theorems & Definitions (22)

• Proposition 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4
• Corollary 3.5
• proof
• Remark 3.1
• Definition 4.1
• Theorem 4.1