Limit theorems for functionals on the facets of stationary random tessellations

We observe stationary random tessellations X = {Ξn}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 motioninvariant tessellations Xn generated within each cell Ξn 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).