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April 2019 Normal approximation for stabilizing functionals
Raphaël Lachièze-Rey, Matthias Schulte, J. E. Yukich
Ann. Appl. Probab. 29(2): 931-993 (April 2019). DOI: 10.1214/18-AAP1405

Abstract

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 $\mathbb{R}^{d}$, including $m$-dimensional Riemannian manifolds, $m\leq d$. 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 $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. 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.

Citation

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Raphaël Lachièze-Rey. Matthias Schulte. J. E. Yukich. "Normal approximation for stabilizing functionals." Ann. Appl. Probab. 29 (2) 931 - 993, April 2019. https://doi.org/10.1214/18-AAP1405

Information

Received: 1 December 2017; Revised: 1 May 2018; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047442
MathSciNet: MR3910021
Digital Object Identifier: 10.1214/18-AAP1405

Subjects:
Primary: 60F05
Secondary: 60D05

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2019
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