Convergence rate for geometric statistics of point processes with fast decay dependence

TL;DR

This paper investigates the convergence rates of normal approximations for geometric statistics of point processes with fast decay dependence, providing explicit error bounds in Wasserstein distance for specific point process models.

Contribution

It extends previous CLT results by quantifying the approximation errors and applies these to Gibbs and determinantal point processes with fast decay kernels.

Findings

Established Wasserstein distance bounds for normal approximation

Applied results to Gibbs and determinantal point processes

Demonstrated the effectiveness of the bounds in specific models

Abstract

[B{\l}aszczyszyn, Yogeshwaran and Yukich (2019)] established central limit theorems for geometric statistics of point processes having fast decay dependence. As limit theorems are of limited use unless we understand their errors involved in the approximation, in this paper, we consider the rates of a normal approximation in terms of the Wasserstein distance for statistics of point processes on $\mathbb{R}^d$ satisfying fast decay dependence. We demonstrate the use of the theorems for statistics arising from two families of point processes: the rarified Gibbs point processes and the determinantal point processes with fast decay kernels.