Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field

TL;DR

This paper establishes Berry-Esseen bounds for the sum of spins in an inhomogeneous Curie-Weiss model with external field, extending Stein's method to a multidimensional setting with unbounded variables.

Contribution

It introduces a generalized Stein's method for multidimensional unbounded variables to derive convergence rates in the inhomogeneous Curie-Weiss model.

Findings

Derived Berry-Esseen bounds for the model's central limit theorem

Extended Stein's method to multidimensional unbounded variables

Quantified convergence rates in the inhomogeneous Curie-Weiss model

Abstract

We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.