Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs

Abstract

An exchangeable pair approach is commonly taken in the normal and non-normal 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 non-normal 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.

51 pages

References in corpus (3)

• A new method of normal approximation
• Exchangeable pairs and Poisson approximation
• A short survey of Stein's method

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• Cramér-type Moderate Deviation Theorems for Nonnormal Approximation
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• Quantitative two-scale stabilization on the Poisson space
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• Normal approximation of Kabanov-Skorohod integrals on Poisson spaces
• Error bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models
• Berry--Esseen bounds for generalized $U$ statistics
• Three-Parameter Approximations of Sums of Locally Dependent Random Variables via Stein's Method
• Non-uniform Berry-Esseen Bound by Unbounded Exchangeable Pair Approach
• Rates of convergence in the CLT for nonlinear statistics under relaxed moment conditions