Variance asymptotics and scaling limits for Gaussian Polytopes

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

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

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input.

Mots clés: parabolic germ-grain models; Random polytopes; convex hulls of Gaussian samples; Poisson point processes; Burgers' equation

Domaines: Probabilités [math.PR]

Pierre Calka, J. E. Yukich. Variance asymptotics and scaling limits for Gaussian Polytopes. Probability Theory and Related Fields, 2015, 163 (1-2), pp.259-301. ⟨hal-00955664v2⟩