Variance Asymptotics and Scaling Limits for Random Polytopes

Pierre Calka (1) , J. E. Yukich (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Lehigh University - Department of Mathematics

Let K be a convex set in R d and let K λ be the convex hull of a homogeneous Poisson point process P λ of intensity λ on K. When K is a simple polytope, we establish scaling limits as λ → ∞ for the boundary of K λ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K λ , k ∈ {0, 1, ..., d − 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K λ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d−1 × R having intensity √ de dh dhdv.

Keywords: stabilization; germ-grain models; Random polytopes

Domains: Probability [math.PR]

Pierre Calka, J. E. Yukich. Variance Asymptotics and Scaling Limits for Random Polytopes. Advances in Mathematics, 2017, 304, pp.1-55. ⟨hal-01263125⟩