## The Annals of Probability

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

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691618

Digital Object Identifier
doi:10.1214/18-AOP1255

Mathematical Reviews number (MathSciNet)
MR3909966

#### Citation

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. https://projecteuclid.org/euclid.aop/1544691618

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