广义Gamma多面体的泛标度极限
Universal Scaling Limits for Generalized Gamma Polytopes
摘要：
固定一个空间维数$d\ge 2$，参数$α> -1$和$β\ge 1$，设$γ_{d,α, β}$为$\mathbb{R}^d$中密度与\begin{align*} ||x||^α\, \exp\left(-\frac{\|x\|^β}β\right), \qquad x\in \mathbb{R}^d. \end{align*}成正比的各向同性随机向量的概率测度。通过$K_λ$，我们将产生于$\mathbb{R}^d$中强度测度为$λγ_{d,α,β}$，$λ>0$的Poisson点过程的随机凸包的广义Gamma多面体表示为。我们建立的缩放限制的边界的$K_λ$，作为$λ\rightarrow \infty$，是由一个普遍的“花彩”分段抛物面，独立的$α$和$β$。此外，我们还列举了其他一些大规模渐近结果，包括期望和方差渐近性，中心极限定理，浓度不等式，Marcinkiewicz-Zygmund型强大数定律，以及$K_λ$的内禀体积和面数的中偏差原理.
英文摘要：
Fix a space dimension $d\ge 2$, parameters $α> -1$ and $β\ge 1$, and let $γ_{d,α, β}$ be the probability measure of an isotropic random vector in $\mathbb{R}^d$ with density proportional to \begin{align*} ||x||^α\, \exp\left(-\frac{\|x\|^β}β\right), \qquad x\in \mathbb{R}^d. \end{align*} By $K_λ$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in $\mathbb{R}^d$ with intensity measure $λγ_{d,α,β}$, $λ>0$. We establish that the scaling limit of the boundary of $K_λ$, as $λ\rightarrow \infty$, is given by a universal `festoon' of piecewise parabolic surfaces, independent of $α$ and $β$. 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 $K_λ$.
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