• Bernoulli
  • Volume 13, Number 3 (2007), 868-891.

Limit theorems for functionals on the facets of stationary random tessellations

Lothar Heinrich, Hendrik Schmidt, and Volker Schmidt

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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≤kd−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).

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Bernoulli, Volume 13, Number 3 (2007), 868-891.

First available in Project Euclid: 7 August 2007

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asymptotic variance β-mixing central limit theorem k-facet process nesting of tessellation Poisson hyperplane process Poisson–Voronoi tessellation weakly dependent tessellation


Heinrich, Lothar; Schmidt, Hendrik; Schmidt, Volker. Limit theorems for functionals on the facets of stationary random tessellations. Bernoulli 13 (2007), no. 3, 868--891. doi:10.3150/07-BEJ6131.

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