Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model

Abstract

Let $(W,W^{\prime })$ be an exchangeable pair. Assume that \[E(W-W'|W)=g(W)+r(W),\] where $g(W)$ is a dominated term and $r(W)$ is negligible. Let $G(t)={\int }_0^{t}g(s)ds$ and define $p(t)=c_1{e}^{-c_{0}G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/{\int }_{-\infty }^{\infty }{e}^{-c_{0}G(t)}dt$. Let $Y$ be a random variable with the probability density function $p$. It is proved that $W$ converges to $Y$ in distribution when the conditional second moment of $(W-W^{\prime })$ given $W$ satisfies a law of large numbers. A Berry-Esseen type bound is also given. We use this technique to obtain a Berry-Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie-Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli-Laplace Markov chain is also discussed.

Published in at http://dx.doi.org/10.1214/10-AAP712 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• New rates for exponential approximation and the theorems of Rényi and Yaglom
• Exponential Approximation by Stein's Method and Spectral Graph Theory

Cited by in corpus (20)

• Local limit theorems via Landau-Kolmogorov inequalities
• Berry-Esseen Bounds of Normal and Non-normal Approximation for Unbounded Exchangeable Pairs
• From Stein identities to moderate deviations
• Stein's method in high dimensions with applications
• Generalized gamma approximation with rates for urns, walks and trees
• Fixed point characterizations of continuous univariate probability distributions and their applications
• Fourth moment theorems on the Poisson space in any dimension
• Fluctuation results for general block spin Ising models
• Asymptotics of mean-field $O(N)$ models
• Phase Transitions of the Maximum Likelihood Estimates in the $p$-Spin Curie-Weiss Model
• Stein's method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process
• Asymptotics of the mean-field Heisenberg model
• Limit theorems for the cubic mean-field Ising model
• Limit theorems for monomer-dimer mean-field models with attractive potential
• Cramér-type Moderate Deviation Theorems for Nonnormal Approximation
• Phase transitions for rates of convergence in the Blume-Emery-Griffiths model
• Stable and metastable phases for the Curie-Weiss-Potts model in vector-valued fields via singularity theory
• Limit Theorems in the Imitative Monomer-Dimer Mean-Field Model via Stein's Method
• Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field
• A surrogate by exchangeability approach to the Curie-Weiss model