Intersections of Randomly Translated Sets

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