Limit theorems for stationary tessellations with random inner cell structures

Abstract

We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.