A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Abstract

We prove large deviation results for Minkowski sums S n of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summand. These results extend those in Mikosch, Pawlas and Samorodnitsky (2011) for generally nonconvex sets, where we assumed that the normalization of S n grows faster than n.