The Extremal Index for a Random Tessellation

Abstract

Let $m$ be a random tessellation in $\mathbf {R}^d$, $d\ge 1$, observed in the window $\mathbf {W}_{\!\rho }=\rho ^{1/d}[0,1]^d$, $\rho >0$, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells $C\in m$ with nucleus in $\mathbf {W}_{\!\rho }$ as $\rho$ goes to infinity. When the normalized maximum converges, we show that its asymptotic distribution depends on the so-called extremal index. Two examples of extremal indices are provided for Poisson-Voronoi and Poisson-Delaunay tessellations.