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).
Publié le : 2007-08-14
Classification:
asymptotic variance,
β-mixing,
central limit theorem,
k-facet process,
nesting of tessellation,
Poisson hyperplane process,
Poisson–Voronoi tessellation,
weakly dependent tessellation
@article{1186503491,
author = {Heinrich, Lothar and Schmidt, Hendrik and Schmidt, Volker},
title = {Limit theorems for functionals on the facets of stationary random tessellations},
journal = {Bernoulli},
volume = {13},
number = {1},
year = {2007},
pages = { 868-891},
language = {en},
url = {http://dml.mathdoc.fr/item/1186503491}
}
Heinrich, Lothar; Schmidt, Hendrik; Schmidt, Volker. Limit theorems for functionals on the facets of stationary random tessellations. Bernoulli, Tome 13 (2007) no. 1, pp. 868-891. http://gdmltest.u-ga.fr/item/1186503491/