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December 2021 Palm theory, random measures and Stein couplings
Louis H. Y. Chen, Adrian Röllin, Aihua Xia
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Ann. Appl. Probab. 31(6): 2881-2923 (December 2021). DOI: 10.1214/21-AAP1666

Abstract

We establish a general Berry–Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre–Voronoi tessellations and of Poisson–Voronoi tessellations. Moreover, we apply the general bound to Stein couplings and discuss the special cases of local dependence and additive functionals in occupancy problems.

Funding Statement

This research was supported by the ARC Discovery Grants DP150101459 and DP190100613, the Singapore Ministry of Education Academic Research Fund Tier 2 Grant MOE2018-T2-2-076, and Singapore Ministry of Education Academic Research Fund Tier 1 Grants R-146-000-182-112 and R-146-000-230-114.

Acknowledgments

We thank the Institute of Mathematical Sciences, NUS, for supporting the workshop Workshop on New Directions in Stein’s Method in March 2015, during which part of this research was conducted.

Citation

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Louis H. Y. Chen. Adrian Röllin. Aihua Xia. "Palm theory, random measures and Stein couplings." Ann. Appl. Probab. 31 (6) 2881 - 2923, December 2021. https://doi.org/10.1214/21-AAP1666

Information

Received: 1 April 2020; Revised: 1 August 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/21-AAP1666

Subjects:
Primary: 60F05
Secondary: 60G55 , 60G57

Keywords: Normal approximation , Palm distribution , random measure , Stein coupling , Stein’s method , Stochastic geometry

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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