Diierentiable Selections of Set-valued Mappings and Asymptotic Behaviour of Random Sets in Innnite Dimensions

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