On the rate of normal approximation for Poisson continuum percolation
Preprint
Abstract
Penrose (2003) states that the number of points in the largest cluster of a percolating Poisson process restricted to a large finite box is asymptotically normal. In this note, we establish a rate of convergence for the statement. As the largest cluster exhibits long-range dependence, we use the fact that the second largest cluster has comparatively shorter range of dependence to restrict the range of dependence, apply Corollary 3.2 of Chen, R\"ollin and Xia (2021) to obtain a Berry-Esseen type bound for the normal approximation of the number of points belonging to clusters that have a restricted range of dependence, and then estimate the gap between this quantity and the number of points in the largest cluster.Keywords
All Related Versions
This publication has 9 references indexed in Scilit:
- Normal approximation in total variation for statistics in geometric probabilityAdvances in Applied Probability, 2023
- Gaussian approximation for sums of region-stabilizing scoresElectronic Journal of Probability, 2022
- Palm theory, random measures and Stein couplingsThe Annals of Applied Probability, 2021
- Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilizationProbability Theory and Related Fields, 2015
- Normal Approximation of Poisson Functionals in Kolmogorov DistanceJournal of Theoretical Probability, 2014
- A central limit theorem for the Poisson–Voronoi approximationAdvances in Applied Mathematics, 2012
- Normal Approximation by Stein’s Method2011
- Continuum percolation with steps in the square or the discRandom Structures & Algorithms, 2005
- Central Limit Theorems for Some Graphs in Computational GeometryThe Annals of Applied Probability, 2001