[Limites non uniformes de Berry–Esseen pour les paires échangeables avec applications aux modèles classiques à
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
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
Révisé le :
Accepté le :
Publié le :
Keywords: Stein’s method, exchangeable pair, Kolmogorov distance, non-uniform bound, mean-field classical
Mots-clés : Méthode de Stein, paire échangeable, distance de Kolmogorov, limite non uniforme, modèle classique à
Lê Vǎn Thành 1 ; Nguyen Ngoc Tu 2

@article{CRMATH_2025__363_G8_757_0, author = {L\^e Vǎn Th\`anh and Nguyen Ngoc Tu}, title = {Non-uniform {Berry{\textendash}Esseen} bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and {Jack} measures}, journal = {Comptes Rendus. Math\'ematique}, pages = {757--776}, publisher = {Acad\'emie des sciences, Paris}, volume = {363}, year = {2025}, doi = {10.5802/crmath.711}, language = {en}, }
TY - JOUR AU - Lê Vǎn Thành AU - Nguyen Ngoc Tu TI - Non-uniform Berry–Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures JO - Comptes Rendus. Mathématique PY - 2025 SP - 757 EP - 776 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.711 LA - en ID - CRMATH_2025__363_G8_757_0 ER -
%0 Journal Article %A Lê Vǎn Thành %A Nguyen Ngoc Tu %T Non-uniform Berry–Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures %J Comptes Rendus. Mathématique %D 2025 %P 757-776 %V 363 %I Académie des sciences, Paris %R 10.5802/crmath.711 %G en %F CRMATH_2025__363_G8_757_0
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/
[1] Mathematical methods for physicists, Elsevier, 2005, xii+1182 pages | MR | Zbl
[2]
[3] Non-uniform Berry–Esseen bounds for Gaussian, Poisson and Rademacher processes (2024) | arXiv | Zbl
[4] On the error bound in a combinatorial central limit theorem, Bernoulli, Volume 21 (2015) no. 1, pp. 335-359 | MR | Zbl
[5] From Stein identities to moderate deviations, Ann. Probab., Volume 41 (2013) no. 1, pp. 262-293 | MR | Zbl
[6] Normal approximation by Stein’s method, Probability and Its Applications, Springer, 2011, xii+405 pages | DOI | MR | Zbl
[7] A generalized Kubilius–Barban–Vinogradov bound for prime multiplicities, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 20 (2023) no. 1, pp. 713-730 | DOI | Zbl
[8] On the error bound in the normal approximation for Jack measures, Bernoulli, Volume 27 (2021) no. 1, pp. 442-468 | MR | Zbl
[9] A non-uniform Berry–Esseen bound via Stein’s method, Probab. Theory Relat. Fields, Volume 120 (2001), pp. 236-254 | DOI | MR | Zbl
[10] Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J., Volume 165 (2016) no. 7, pp. 1193-1282 | MR | Zbl
[11] Stein’s method and a cubic mean-field model, J. Stat. Phys., Volume 191 (2024) no. 12, 163, 27 pages | DOI | MR | Zbl
[12] Stein’s method for dependent random variables occuring in statistical mechanics, Electron. J. Probab., Volume 15 (2010) no. 30, pp. 962-988 | MR | Zbl
[13] A refined Cramér-type moderate deviation for sums of local statistics, Bernoulli, Volume 26 (2020) no. 3, pp. 2319-2352 | MR | Zbl
[14] Statistical mechanics of lattice systems. A concrete mathematical introduction, Cambridge University Press, 2018, xix+622 pages | MR | Zbl
[15] Stein’s method, Jack measure, and the Metropolis algorithm, J. Comb. Theory, Ser. A, Volume 108 (2004) no. 2, pp. 275-296 | DOI | MR | Zbl
[16] An inductive proof of the Berry–Esseen theorem for character ratios, Ann. Comb., Volume 10 (2006), pp. 319-332 | DOI | MR | Zbl
[17] Zero biasing and Jack measures, Comb. Probab. Comput., Volume 20 (2011) no. 5, pp. 753-762 | DOI | MR | Zbl
[18] Rigidity and edge universality of discrete
[19] Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein’s method for distributional approximations, Math. Proc. Camb. Philos. Soc., Volume 147 (2009) no. 1, pp. 95-114 | DOI | MR | Zbl
[20] Law of large numbers and central limit theorems through Jack generating functions, Adv. Math., Volume 380 (2021), 107545, 91 pages | MR | Zbl
[21] Anisotropic Young diagrams and Jack symmetric functions, Funct. Anal. Appl., Volume 34 (2000), pp. 41-51 | DOI | MR | Zbl
[22] Asymptotics of mean-field
[23] Quantitative normal approximation of linear statistics of
[24] Cramér-type moderate deviations under local dependence, Ann. Appl. Probab., Volume 33 (2023) no. 6A, pp. 4747-4797 | MR | Zbl
[25] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, 1995, x+475 pages | DOI | MR | Zbl
[26] The uses of random partitions, XIVth International Congress on Mathematical Physics, World Scientific (2006), pp. 379-403 | DOI | Zbl
[27] Degree asymptotics with rates for preferential attachment random graphs, Ann. Appl. Probab., Volume 23 (2013) no. 3, pp. 1188-1218 | MR | Zbl
[28] The Berry–Esseen bound for character ratios, Proc. Am. Math. Soc., Volume 134 (2006) no. 7, pp. 2153-2159 | DOI | MR | Zbl
[29] Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs, Ann. Probab., Volume 47 (2019) no. 1, pp. 61-108 | MR | Zbl
[30] Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes - Monograph Series, 7, Institute of Mathematical Statistics, 1986, iv+164 pages | DOI | MR | Zbl
[31] A moment inequality for exchangeable pairs with applications (2025), pp. 1-12 (Manuscript in preparation)
[32] Error bounds in normal approximation for the squared-length of total spin in the mean field classical
[33] Non-uniform bounds for non-normal approximation via Stein’s method with applications to the Curie–Weiss model and the imitative monomer-dimer model (2025) | arXiv
[34] Non-uniform Berry–Esseen bounds via Malliavin–Stein method, C. R. Math., Volume 363 (2025), pp. 455-463 | DOI | MR | Zbl
Cité par Sources :
Commentaires - Politique