# Error bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models

## TL;DR

This paper establishes new Kolmogorov bounds and improved Wasserstein bounds for the normal approximation of the squared-length of total spin in mean field classical N-vector models, using Stein's method.

## Contribution

It introduces a new Kolmogorov bound and enhances the Wasserstein bound for the normal approximation in these models, advancing existing results.

## Key findings

- Kolmogorov bound is newly derived.
- Wasserstein bound is improved over previous work.
- Results apply Stein's method to exchangeable pairs in spin models.

## Abstract

This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, \textbf{165} (2016), no. 6, 1114--1140]. The proof is based on Stein's method for exchangeable pairs.