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

TL;DR

The paper proves a sharp non-uniform Berry-Esseen bound for normal approximation of exchangeable pairs under a nonlinear regression condition with unbounded increments, explicitly depending on moments of W up to order 2p. Using Stein's method together with a non-uniform concentration inequality, it bounds the error by a decaying (1+|z|)^{-p} factor multiplied by terms involving the conditional variance deviation, the remainder R, a truncation parameter a, and a high-moment tail of Δ. The main result is then applied to two dependent-structure settings: the squared-length of the total spin in mean-field N-vector models and Jack deformations of the character ratio, yielding improved non-uniform bounds (rates of order n^{-1/2}) over previous uniform results and broadening the applicability of Stein’s method for dependent data.

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 Löwe [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.

Theorems & Definitions (17)

• Theorem 1.1: Eichelsbacher eichelsbacher2024stein
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Theorem \ref{['thm.main11']}
• Theorem 3.1
• proof
• Theorem 3.2