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January 2019 Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs
Qi-Man Shao, Zhuo-Song Zhang
Ann. Probab. 47(1): 61-108 (January 2019). DOI: 10.1214/18-AOP1255


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.


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Qi-Man Shao. Zhuo-Song Zhang. "Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs." Ann. Probab. 47 (1) 61 - 108, January 2019.


Received: 1 November 2016; Revised: 1 December 2017; Published: January 2019
First available in Project Euclid: 13 December 2018

zbMATH: 07036334
MathSciNet: MR3909966
Digital Object Identifier: 10.1214/18-AOP1255

Primary: 60F05
Secondary: 60K35

Keywords: Berry–Esseen bound , Exchangeable pairs , general Curie–Weiss model , mean field Heisenberg model , monochromatic edges , Quadratic forms , simple random sampling , Stein’s method

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 1 • January 2019
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