A general study of extremes of stationary tessellations with applications

Let $\mathfrak{m}$ be a random tessellation in $\mathbf{R}^d$ observed in a bounded Borel subset $W$ and $f(\cdot)$ be a measurable function defined on the set of convex bodies. To each cell $C$ of $\mathfrak{m}$ we associate a point $z(C)$ which is the nucleus of $C$. Applying $f(\cdot)$ to all the cells of $\mathfrak{m}$, we investigate the order statistics of $f(C)$ over all cells $C\in\mathfrak{m}$ with nucleus in $\mathbf{W}_ρ=ρ^{1/d}W$ when $ρ$ goes to infinity. Under a strong mixing property and a local condition on $\mathfrak{m}$ and $f(\cdot)$, we show a general theorem which reduces the study of the order statistics to the random variable $f(\mathscr{C})$ where $\mathscr{C}$ is the typical cell of $\mathfrak{m}$. The proof is deduced from a Poisson approximation on a dependency graph via the Chen-Stein method. We obtain that the point process $\left\{(ρ^{-1/d}z(C), a_ρ^{-1}(f(C)-b_ρ)), C\in\mathfrak{m}, z(C)\in \mathbf{W}_ρ\right\}$, where $a_ρ>0$ and $b_ρ$ 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 $f(\cdot)$ such as the inradius, the circumradius, the area, the volume of the Voronoi flower and the distance to the farthest neighbor. When the local condition does not hold and the normalized maximum converges, the asymptotic behaviour depends on two quantities that are the distribution function of $f(\mathscr{C})$ and a constant $θ\in [0,1]$ which is the so-called extremal index.

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