A general study of extremes of stationary tessellations with examples

Let m be a random tessellation in Rd, d≥1, observed in a bounded Borel subset W and f(⋅) be a measurable function defined on the set of convex bodies. A point z(C), called the nucleus of C, is associated with each cell C of m. Applying f(⋅) to all the cells of m, we investigate the order statistics of f(C) over all cells C∈m with nucleus in Wρ=ρ1/dW when ρ goes to infinity. Under a strong mixing property and a local condition on m and f(⋅), we show a general theorem which reduces the study of the order statistics to the random variable f(C), where C is the typical cell of m. The proof is deduced from a Poisson approximation on a dependency graph via the Chen–Stein method. We obtain that the point process {(ρ−1/dz(C),aρ−1(f(C)−bρ)),C∈m,z(C)∈Wρ}, where aρ>0 and bρ are two suitable functions depending on ρ, converges to a non-homogeneous Poisson point process. Several applications of the general theorem are derived in the particular setting of Poisson–Voronoi and Poisson–Delaunay tessellations and for different functions f(⋅) such as the inradius, the circumradius, the area, the volume of the Voronoi flower and the distance to the farthest neighbor.