Non-topological large deviations via maxitive monetary risk measures

In large deviations theory one usually has a family of probability measures in some space $X$ with a topology. In this paper we study what kind of large deviation results can be established when no topology is assumed on the underlying measure space $X$. The main tool is duality theory of risk measures. We prove that a monetary risk measures $\phi$ satisfies the Laplace principle $\phi(f)={\rm ess.sup}_X(f-I)$ for some measurable function $I\colon X\to[0,\infty]$ if and only $\phi$ is maxitive and continuous from below. In the special case of the asymptotic entropy $\psi(f)=\lim_{n\to\infty} \tfrac{1}{n}\log \int e^{n f} d\nu_n$, this result can been understood as a non-topological analogue of the Varadhan's integral lemma. In the spirit of Bryc's theorem, we study non-topological analogues of exponential tightness which ensure the Laplace principle characterised in the main result. The theory is applied to obtain non-topological large deviation results for sequences of sublinear expectations.