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

TL;DR

This paper develops a nonnormal approximation method using Stein's exchangeable pairs technique, applying it to the Curie-Weiss model to derive error bounds and analyze spectral properties of Markov chains.

Contribution

It introduces a novel Stein's method approach for nonnormal approximation with exchangeable pairs, applied to the Curie-Weiss model and Markov chain spectra.

Findings

Established a convergence in distribution to a nonnormal limit for the Curie-Weiss magnetization.

Derived a Berry-Esseen bound of order 1/√n for the noncentral limit theorem.

Discussed exponential approximation for the spectrum of Bernoulli-Laplace chains.

Abstract

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.