The Annals of Probability

Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs

Qi-Man Shao and Zhuo-Song Zhang

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An exchangeable pair approach is commonly taken in the normal and nonnormal approximation using Stein’s method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein’s method, a new Berry–Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and nonnormal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie–Weiss model, mean field Heisenberg model and colored graph model.

Article information

Ann. Probab., Volume 47, Number 1 (2019), 61-108.

Received: November 2016
Revised: December 2017
First available in Project Euclid: 13 December 2018

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Mathematical Reviews number (MathSciNet)

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Stein’s method exchangeable pairs Berry–Esseen bound quadratic forms simple random sampling general Curie–Weiss model mean field Heisenberg model monochromatic edges


Shao, Qi-Man; Zhang, Zhuo-Song. Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs. Ann. Probab. 47 (2019), no. 1, 61--108. doi:10.1214/18-AOP1255.

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