Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Brownian Limits, Local Limits and Variance Asymptotics for Convex Hulls in the Ball by Pierre Calka,

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball Bd , d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the par...

Brownian limits, local limits, extreme value and variance asymptotics for convex hulls in the ball

The paper [40] establishes an asymptotic representation for random convex polytope geometry in the unit ball Bd, d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of the so-called generalized paraboloid growth process. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via ...

Variance Asymptotics and Scaling Limits for Random Polytopes

Let K be a convex set in Rd and let Kλ be the convex hull of a homogeneous Poisson point process Pλ of intensity λ on K. When K is a simple polytope, we establish scaling limits as λ → ∞ for the boundary of Kλ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of Kλ, k ∈ {0, 1, ..., d − 1}, resolving an open question posed in [18]. The scaling l...

Variance Asymptotics and Scaling Limits for Gaussian Polytopes

Let Kn be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on Rd. We establish variance asymptotics for the re-scaled volume and k-face functional of Kn, k ∈ {0, 1, ..., d − 1}, resolving an open problem. Asymptotic variances and the scaling limit of the boundary of Kn are given in terms of functionals of germ-grain models having parabolic gra...

Generalized Covering Groups and Direct Limits