Large and moderate deviations for random sets and random upper semicontinuous functions

Abstract

Firstly, we obtain sample path large deviations for compact random sets, the main tool is a result of large deviations on D([0,1],B) with the uniform metric. We also show that Cerf’s result (1999) [5] is only a corollary of sample path large deviations. Secondly, we obtain large deviations and moderate deviations of random sets which take values of bounded closed convex sets on the underling separable Banach space with respect to the Hausdorff distance d_H and that of random upper semicontinuous functions whose values are of bounded closed convex levels on the underling separable Banach space in the sense of the uniform Hausdorff distance $d_H^{\infty }\text{.}$ The main tool is the work of Wu on the large deviations and moderate deviations for empirical processes (Wu, 1994) [27]. Finally, we prove that Lemma 2 in [5], which is very important for “deconvexification”, still holds under another condition $E[\exp \{\lambda \Vert X{\Vert }_{\mathcal{K}}^p\}]<\infty$ for some $\lambda >0$ in a different proof method.