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

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.