Abstract
We observe stationary random tessellations X={Ξn}n≥1 in ℝd 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 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).
Citation
Lothar Heinrich. Hendrik Schmidt. Volker Schmidt. "Limit theorems for functionals on the facets of stationary random tessellations." Bernoulli 13 (3) 868 - 891, August 2007. https://doi.org/10.3150/07-BEJ6131
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