Abstract.
Consider a sequence of stationary tessellations {Θn}, n=0,1,…, of ℝd consisting of cells {C n(x i n)}with the nuclei {x i n}. An aggregate cell of level one, C 0 1(x i 0), is the result of merging the cells of Θ1 whose nuclei lie in C 0(x i 0). An aggregate tessellation Θ0 n consists of the aggregate cells of level n, C 0 n(x i 0), defined recursively by merging those cells of Θn whose nuclei lie in C n −1(x i 0).
We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as n→∞ and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {Θn}.
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Received: 3 June 1999 / Revised version: 22 November 2000 / Published online: 24 July 2001
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Tchoumatchenko, K., Zuyev, S. Aggregate and fractal tessellations. Probab Theory Relat Fields 121, 198–218 (2001). https://doi.org/10.1007/PL00008802
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DOI: https://doi.org/10.1007/PL00008802