Weakly maxitive set functions and their possibility distributions

TL;DR

This paper explores a weaker form of maxitivity in set functions, demonstrating that under certain conditions, the associated maxitive integrals are still uniquely determined by possibility distributions, with applications in large deviations theory.

Contribution

It introduces a weaker topological maxitivity concept, extending the characterization of maxitive integrals and providing new results in large deviations and non-linear expectations.

Findings

Maxitive integrals are determined by possibility distributions under weaker assumptions.

A Laplace principle for maxitive integrals is established.

Large deviations bounds and a monotone Cramér's theorem are derived.

Abstract

The Shilkret integral with respect to a completely maxitive capacity is fully determined by a possibility distribution. In this paper, we introduce a weaker topological form of maxitivity and show that under this assumption the Shilkret integral is still determined by its possibility distribution for functions that are sufficiently regular. Motivated by large deviations theory, we provide a Laplace principle for maxitive integrals and characterize the possibility distribution under certain separation and convexity assumptions. Moreover, we show a maxitive integral representation result for weakly maxitive non-linear expectations. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities, and by deriving a monotone analogue of Cram\'{e}r's theorem.