Differentiable Selections of Set-Valued Mappings and Asymptotic Behavior of Random Sets in Infinite Dimensions

Abstract

We consider set-valued mappings acting between two linear normed spaces and having convex closed images. Our aim is to construct selections with directional differentiability properties up to the second order, using certain tangential approximations of the mapping. The constructions preserve measurability and lead to a directionally differentiable Castaing representation of measurable multifunctions admitting the required tangential approximation. A generalized set-valued central limit theorem for random sets in infinite dimensional spaces is presented. The results yield asymptotic distributions of measurable selections forming the Castaing representation of the multifunction.