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Normal approximation in total variation for statistics in geometric probability

Part of: Limit theorems Geometric probability and stochastic geometry Distribution theory Stochastic processes

Published online by Cambridge University Press:  03 July 2023

Tianshu Cong Open the ORCID record for Tianshu Cong [Opens in a new window]  and
Aihua Xia
Tianshu Cong*
Affiliation:
Jilin University, University of Melbourne
Aihua Xia*
Affiliation:
University of Melbourne
*
*Postal address: School of Mathematics and Statistics, the University of Melbourne, Parkville VIC 3010, Australia.
*Postal address: School of Mathematics and Statistics, the University of Melbourne, Parkville VIC 3010, Australia.
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Abstract

We use Stein’s method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of samples arising from random events driven by a marked Poisson point process on $\mathbb{R}^d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilisation and moment conditions. At the cost of an additional non-singularity condition, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellations, k-nearest-neighbours graphs, timber volume, and maximal layers.

Keywords

Total variation distancenon-singular distributionBerry–Esseen boundStein’s method

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 62E20: Asymptotic distribution theory
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 1 , March 2024 , pp. 106 - 155
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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