Maxitive integration and its application to monotone large deviations.

In this paper, we provide duality bounds for the maxitive integral of a non-additive set function. These bounds allow for a monotone analogue of the basic results in large deviations theory. In particular, we show the equivalence of the large deviation principle and the Laplace principle for general concentration functions and characterize the respective monotone rate function. As an application of the results, we obtain on the one hand a monotone version of Cram\'{e}r's theorem, and on the other hand a large deviation and Laplace principle for the asymptotic comparison of random walks as introduced by Fritz [8].