Higher moments of intrinsic volumes of random beta-prime polytopes

Abstract

We consider beta-prime polytopes, i.e., the convex hulls of iid random points chosen according to beta-prime distributions in ${\mathbb{R}}^d$. After suitable scaling, beta-prime polytopes converge in distribution to the convex hulls of Poisson point processes with power-law intensity functions. We prove moment convergence for the volume and all intrinsic volumes. Beta-prime polytopes are the push-forwards of spherical random polytopes on the upper open half-sphere of the unit sphere $S^d\subset {\mathbb{R}}^{d+1}$. We prove convergence of moments of the spherical volume difference of the half-sphere and the spherical random polytopes.

Major changes in results. The variance lower bound results are replaced with convergence of moments. The arguments are completely different from those in the previous version