Intersections of randomly translated sets

Let ${\mathrm{\Xi }}_n=\{{\xi }_1,\dots ,{\xi }_n\}$ be a sample of $n$ independent points distributed in a regular closed element $K$ of the extended convex ring in ${\mathbb{R}}^d$ according to a probability measure $\mu$ on $K$, admitting a density function. We consider random sets generated from the intersection of the translations of $K$ by elements of ${\mathrm{\Xi }}_n$, as $X_n={\bigcap }_{i=1}^n(K-{\xi }_i)$. This work aims to show that the scaled closure of the complement of $X_n$ as $n\to \mathrm{\infty }$ converges in distribution to the closure of the complement zero cell of a Poisson hyperplane tessellation whose distribution is determined by the curvature measure of $K$ and the behaviour of the density of $\mu$ near the boundary of $K$.