论文标题
双曲线随机几何形状中的大型邻居球
Large nearest neighbour balls in hyperbolic stochastic geometry
论文作者
论文摘要
考虑在$ d $维的双曲线空间中的固定泊松过程。对于$ r> 0 $，定义点过程$ξ_r^{（k）} $超高高度的$ th $ k $ th的合适阈值，该量的$ k $ th最接近的邻居球以poisson过程的点为中心，围绕着poisson过程的点，以$ r $ $ r $ $ r $的重点为中心。显示了点过程$ξ_r^{（k）} $与kantorovich-rubinstein距离中的“强度函数” $ e^{ - u} $与真实线上的不均匀泊松过程进行了比较。由此，得出了具有限制性牙龈分布的双曲线最大$ k $ th最近的邻居球的定量限制定理。
Consider a stationary Poisson process in a $d$-dimensional hyperbolic space. For $R>0$ define the point process $ξ_R^{(k)}$ of exceedance heights over a suitable threshold of the hyperbolic volumes of $k$th nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius $R$ centred at a fixed point. The point process $ξ_R^{(k)}$ is compared to an inhomogeneous Poisson process on the real line with intensity function $e^{-u}$ and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum $k$th nearest neighbour ball with a limiting Gumbel distribution is derived.
