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)