Intersections of Randomly Translated Sets

Let \(\Xi _n=\{\xi _1,\dots ,\xi _n\}\) be a sample of n independent points distributed on 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 \(\Xi _n\) , as \(\begin{aligned} X_n=\bigcap _{i=1}^n (K-\xi _i). \end{aligned}\) This work aims to show that scaled \(X_n\) as \(n\rightarrow \infty \) converges in distribution to the 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.