Non-uniform Berry--Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures

Abstract: This paper establishes a non-uniform Berry--Esseen bound in normal approximation for exchangeable pairs using Stein's method via a concentration inequality approach. The main theorem extends and improves several results in the literature, including those of Eichelsbacher and L\"{o}we [Electron. J. Probab. 15, 2010, 962--988], and Eichelsbacher [arXiv:2404.07587, 2024]. The result is applied to obtain a non-uniform Berry--Esseen bound for the squared-length of the total spin in the mean-field classical $N$-vector models, and a non-uniform Berry--Esseen bound for Jack deformations of the character ratio.