We extend the well known large deviation upper bound for sums of independent, identically distributed random variables in IR d by weakening the requirement that the rate function have compact level sets (the classical Cram er condition). To do so we establish an apparently new theorem on approximation of closed convex sets by polytopes.

Using the theory of random closed sets, we extend the statistical framework introduced by Schreiber (11) for inference based on set-valued observations from the case of finite sample spaces to compact metric spaces with continuous distributions .