Variance asymptotics and scaling limits for Gaussian Polytopes

Abstract

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on ${\mathbb{R}}^d$. 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 ${\mathbb{R}}^{d-1}\times \mathbb{R}$ with intensity $e^{h}dhdv$. 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.

References in corpus (1)

• Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points