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Non-uniform Berry–Esseen bounds for exchangeable pairs with applications to the mean-field classical N-vector models and Jack measures
[Limites non uniformes de Berry–Esseen pour les paires échangeables avec applications aux modèles classiques à N vecteurs et aux mesures de Jack]
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 757-776.

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 Löwe [Electron. J. Probab. 15 (2010), 962–988], and Eichelsbacher [J. Stat. Phys. 191 (2024), no. 12, article no. 163 (27 pages)]. 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.

Ce document établit une borne de Berry–Esseen non uniforme en approximation normale pour les paires échangeables en utilisant la méthode de Stein via une approche d’inégalité de concentration. Le théorème principal étend et améliore plusieurs résultats de la littérature, y compris ceux d’Eichelsbacher et Löwe [Electron. J. Probab. 15 (2010), 962–988], et d’Eichelsbacher [J. Stat. Phys. 191 (2024), no. 12, article no. 163 (27 pages)]. Le résultat est appliqué pour obtenir une borne de Berry–Esseen non-uniforme pour la longueur carrée du spin total dans les modèles classiques à N vecteurs en champ moyen, et une borne de Berry–Esseen non-uniforme pour les déformations de Jack du rapport de caractère.

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DOI : 10.5802/crmath.711
Classification : 60F15
Keywords: Stein’s method, exchangeable pair, Kolmogorov distance, non-uniform bound, mean-field classical N-vector model, Jack measure
Mots-clés : Méthode de Stein, paire échangeable, distance de Kolmogorov, limite non uniforme, modèle classique à N vecteurs à champ moyen, mesure de Jack

Lê Vǎn Thành 1 ; Nguyen Ngoc Tu 2

1 Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Viet Nam
2 Department of Applied Sciences, HCMC University of Technology and Education, 01 Vo Van Ngan, Ho Chi Minh City, Viet Nam
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Lê Vǎn Thành; Nguyen Ngoc Tu. Non-uniform Berry–Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 757-776. doi : 10.5802/crmath.711. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.711/

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