The extremal index for a random tessellation

Abstract

Let m be a random tessellation in Rd, d ≥ 1, observed in the window Wρ = ρ1/d[0, 1]d, ρ > 0, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells C ∈ m with nucleus in Wρ as ρ 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.