Limit theorems for functionals on the facets of stationary random tessellations

Abstract

We observe stationary random tessellations X={Ξ_n}_n≥1 in ℝ^d through a convex sampling window W that expands unboundedly and we determine the total (k−1)-volume of those (k−1)-dimensional manifold processes which are induced on the k-facets of X (1≤k≤d−1) by their intersections with the (d−1)-facets of independent and identically distributed motion-invariant tessellations X_n generated within each cell Ξ_n of X. The cases of X being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in W are approximately normally distributed when W is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson–Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).