The Annals of Applied Probability

Normal approximation for stabilizing functionals

Raphaël Lachièze-Rey, Matthias Schulte, and J. E. Yukich

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 931-993.

Dates
Received: December 2017
Revised: May 2018
First available in Project Euclid: 24 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1548298934

Digital Object Identifier
doi:10.1214/18-AAP1405

Mathematical Reviews number (MathSciNet)
MR3910021

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Stein’s method Malliavin calculus stabilization random Euclidean graphs statistics of data sets statistics of convex hulls Voronoi set approximation maximal points

Citation

Lachièze-Rey, Raphaël; Schulte, Matthias; Yukich, J. E. Normal approximation for stabilizing functionals. Ann. Appl. Probab. 29 (2019), no. 2, 931--993. doi:10.1214/18-AAP1405. https://projecteuclid.org/euclid.aoap/1548298934


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