Large Deviation Principles for Non-Additive Set Functions

This paper explores the challenging yet crucial extension of Large Deviation Principles (LDPs) to the domain of non-additive set functions, a class of measures that includes capacities, fuzzy measures, and belief functions. Traditional LDPs, foundational in probability theory and statistical mechanics, quantify the exponential decay rates of probabilities of rare events for sequences of random variables governed by additive probability measures. However, many real-world phenomena, particularly in decision theory, artificial intelligence, risk assessment, and game theory, are better modeled by non-additive measures, which inherently capture ambiguity, uncertainty, and interactions not reducible to simple additivity. We define a framework for constructing empirical non-additive set functions from sequences of observations and investigate the conditions under which these empirical functions satisfy a large deviation principle. Our methodology leverages concepts from convex analysis, Choquet integral theory, and variational representations, adapting them to account for the super- or sub-additivity inherent in these functions. We demonstrate the existence and properties of a rate function for specific classes of non-additive set functions, offering insights into the asymptotic behavior of systems where uncertainty is epistemic or structural, rather than purely stochastic. This work lays theoretical groundwork for analyzing rare events and fluctuations in complex systems beyond the classical additive probability paradigm, opening new avenues for understanding robustness and sensitivity in non-additive environments.