Large deviations built on max-stability

Large deviations theory studies the asymptotic behavior of tails of sequences of probability distributions by means of the Large Deviation Principle (LDP) and the Laplace Principle (LP). The present work is aimed to provide a functional analytic foundation for large deviations built on max-stable monetary risk measures. We introduce the LDP for monetary risk measures and establish the Varadhan-Bryc equivalence between the LDP and the LP by showing that a max-stable monetary risk measure satisfies the LDP if and only if it has a representation in terms of the LP. We prove an analogue of Bryc's lemma for monetary risk measures that are locally max-stable on compact subsets, by establishing two sufficient conditions for the LDP: one is a version of exponential tightness, and the other one covers the case when the rate function does not necessarily have compact sublevel sets. The main results are illustrated by the asymptotic shortfall risk of sequences of random variables.