Normal approximation for stabilizing functionals
Normal approximation for stabilizing functionals
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稳定泛函的正态近似
DOI：
--
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Yukich
中科院分区：
文献类型：
--
作者：
R. Lachièze;Matthias Schulte;J. Yukich
我们建立了一般度量空间上一大类标记Poisson点过程和二项式点过程的几何泛函关于Kolmogorov距离的最优正规收敛速度。当几何泛函可以表示为满足矩条件的指数稳定分数函数的和时，这些比率是有效的。通过将稳定化方法结合到Malliavin-Stein理论中，我们得到了稳定化分数函数和的正态逼近速度，这些和要么是现有速度的改进，要么是此类函数的第一次正态逼近速度。在比ℝd的全维子集更一般的空间上，包括m维黎曼流形m≤d上，我们的泛函的一般收敛速度是成立的。我们利用一般结果推导出随机几何中的几个泛函的改进的和新的法向收敛速度，包括那些其方差随基础集的体积或表面积而重新缩放的泛函。特别地，我们改进了d-≥2维光滑凸体中Poisson和二项随机样本的凸壳的k-面和第i个本征体积泛函的正态收敛速度。我们还给出了最近邻图和高维数据集的统计量、随机样本的极大值点数、基于Voronoi网格的集合逼近中的表面积和体积的估计以及广义随机几何图中的团数的改进的正态收敛速度。
We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of ℝd, including m-dimensional Riemannian manifolds, m≤d. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the k-face and ith intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension d≥2. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.
随机多胞形：内在体积的中心极限定理
DOI：
10.1090/proc/14000
发表时间：
2018
期刊：
arXiv: Metric Geometry
影响因子：
--
作者：
C. Thale;N. Turchi;F. Wespi
通讯作者：
F. Wespi
影响因子：
1.5
作者：
J. Grote;C. Thale
通讯作者：
C. Thale