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

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

2009-07-25

arxiv.org/abs/0907.4450v3

Annals of Applied Probability 2011, Vol. 21, No. 2, 464-483

Mathematics

Probability

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

Scientific paper

10.1214/10-AAP712

Let $(W,W')$ 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 $G(t)=\int_0^tg(s)\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt$. 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 $(W-W')$ 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.

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