On the rate of normal approximation for Poisson continuum percolation

TL;DR

This paper establishes a rate of convergence for the normal approximation of the largest cluster size in Poisson continuum percolation, addressing long-range dependence issues with a novel approach.

Contribution

It introduces a method to quantify the convergence rate for the largest cluster size in Poisson percolation, overcoming challenges from long-range dependencies.

Findings

Provides a Berry-Esseen type bound for the normal approximation

Quantifies the convergence rate for the largest cluster size

Addresses long-range dependence in Poisson percolation

Abstract

It is known 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 each point in the largest cluster is determined by points as far as the diameter of the box, known results in the literature of normal approximation for Poisson functionals cannot be directly applied. To disentangle the long-range dependence of the largest cluster, we use the fact that the second largest cluster has comparatively shorter range of dependence to restrict the range of dependence, apply a recently established result in 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.