Differentiation of sets - The general case

Estate V. Khmaladze, Wolfgang Weil

Published 2013-09-19Version 1

In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limit theorems were established for local empirical processes near the boundary of compact convex sets $K$ in $\R$. The limit processes were shown to live on the normal cylinder $\Sigma$ of $K$, respectively on a class of set-valued derivatives in $\Sigma$. The latter result was based on the concept of differentiation of sets at the boundary $\partial K$ of $K$, which was developed in Khmaladze (2007). Here, we extend the theory of set-valued derivatives to boundaries $\partial F$ of rather general closed sets $F\subset \R$, making use of a local Steiner formula for closed sets, established in Hug, Last and Weil (2004).