Universal Scaling Limits for Generalized Gamma Polytopes

TL;DR

This paper studies the asymptotic geometric properties of generalized gamma polytopes formed from Poisson point processes, revealing a universal boundary shape and establishing various probabilistic limit theorems.

Contribution

It introduces a universal scaling limit for the boundary of generalized gamma polytopes, independent of parameters, and provides comprehensive asymptotic results for their geometric features.

Findings

Boundary converges to a universal 'festoon' of parabolic surfaces

Established expectation and variance asymptotics for intrinsic volumes

Proved central limit theorems and concentration inequalities

Abstract

Fix a space dimension $d\ge 2$, parameters $\alpha > -1$ and $\beta \ge 1$, and let $\gamma_{d,\alpha, \beta}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*} ||x||^\alpha\, \exp\left(-\frac{\|x\|^\beta}{\beta}\right), \qquad x\in \mathbb{R}^d. \end{align*} By $K_\lambda$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in $\mathbb{R}^d$ with intensity measure $\lambda\gamma_{d,\alpha,\beta}$, $\lambda>0$. We establish that the scaling limit of the boundary of $K_\lambda$, as $\lambda \rightarrow \infty$, is given by a universal `festoon' of piecewise parabolic surfaces, independent of $\alpha$ and $\beta$. Moreover, we state a list of other large scale asymptotic results, including expectation and variance asymptotics, central limit theorems, concentration inequalities, Marcinkiewicz-Zygmund-type strong laws of large numbers, as well as moderate deviation principles for the intrinsic volumes and face numbers of $K_\lambda$.