Variance asymptotics and scaling limits for random polytopes

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• 作者：Pierre Calka^a ; ¹ ; ^{pierre.calka@univ-rouen.fr" class="auth_mail" title="E-mail the corresponding author} ; J.E. Yukich^b ; ² ; ^{joseph.yukich@lehigh.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 60F05 ; 52A20 ; secondary ; 60D05 ; 52A23
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：1-55
• 全文大小：1108 K

文摘

Let K   be a convex set in 0f3637a0147e278a80a82de4cf0" title="Click to view the MathML source">R^d${\mathbb{R}}^d$ and let K_λ$K_{\lambda }$ be the convex hull of a homogeneous Poisson point process P_λ${\mathcal{P}}_{\lambda }$ of intensity λ on K. When K   is a simple polytope, we establish scaling limits as λ→∞$\lambda \to \infty$ for the boundary of K_λ$K_{\lambda }$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k  -face functional of K_λ$K_{\lambda }$, k∈{0,1,...,d−1}$k\in \{0,1,...,d-1\}$, resolving an open question posed in [17]. The scaling limit of the boundary of K_λ$K_{\lambda }$ and the variance asymptotics are described in terms of a germ–grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R^d−1×R${\mathbb{R}}^{d-1}\times \mathbb{R}$ having intensity 0fec06fefe16dfe7ba1445391ba">$\sqrt{d}{e}^{dh}dhdv$.