Intersections of Randomly Translated Sets

Let \(\Xi _n=\{\xi _1,\dots ,\xi _n\}\) ${\mathrm{\Xi }}_n=\{{\xi }_1,\cdots ,{\xi }_n\}$ be a sample of n independent points distributed in a regular closed element K of the extended convex ring in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ according to a probability measure \(\mu \) $\mu$ on k admitting a density function. We consider random sets generated from the intersection of the translations of K by the elements of \(\Xi _n\) ${\mathrm{\Xi }}_n$ , namely, \(\begin{aligned} X_n=\bigcap _{i=1}^n (K-\xi _i). \end{aligned}\) $$\begin{array}{c}\hfill X_n=\bigcap _{i=1}^n(K-{\xi }_i).\end{array}$$ This work aims to show that the scaled closure of the complement of \(X_n\) $X_n$ as \(n\rightarrow \infty \) $n\to \infty$ 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 \(\mu \) $\mu$ near the boundary of K.