Non-uniform bounds for non-normal approximation via Stein's method with applications to the Curie--Weiss model and the imitative monomer-dimer model

TL;DR

This paper develops non-uniform Berry--Esseen bounds for non-normal approximations using Stein's method and applies these results to analyze magnetization in the Curie--Weiss and imitative monomer-dimer models, extending existing theorems.

Contribution

It generalizes Stein's method for non-normal approximation to non-uniform bounds and applies it to complex models, extending prior theoretical results.

Findings

Derived non-uniform Berry--Esseen bounds for non-normal limits

Extended results to the Curie--Weiss and monomer-dimer models

Improved understanding of non-central limit theorems in statistical physics

Abstract

This paper establishes a non-uniform Berry--Esseen bound for non-normal approximation using Stein's method. The main theorem generalizes the result of the authors in [Comptes Rendus Mathematique, 2024] to the context of non-normal approximation. As an application of the main result, we derive non-uniform Berry--Esseen bounds in non-central limit theorems for the magnetization in the Curie--Weiss model and the imitative monomer-dimer model. These extend some existing results in the literature, including Theorem 2.1 of Chatterjee and Shao [Ann. Appl. Probab., 2011] and Theorem 1 of Chen [J. Math. Physics, 2016].