Stein交换对方法的非正态近似及其在Curie-Weiss模型中的应用
Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model
摘要：
令$(W,W')$为可交换对。假设\[E(W-W'|W)=g(W)+r(W),\]，其中$g(W)$是支配项，而$r(W)$可以忽略。设$G(t)=\int_0^tg(s)\,ds$并定义$p(t)=c_1e^{-c_0G(t)}$，其中$c_0$是适当选择的常数，而$c_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt$。假设$Y$是具有概率密度函数$p$的随机变量。证明了当给定$W$的$(W-W')$的条件二阶矩满足大数定律时，$W$按分布收敛于$Y$.一个Berry-Esseen型界也给出了。我们利用这种方法得到了居里-外斯铁磁体临界温度下磁化强度的非中心极限定理中的$1/\sqrt{n}$阶Berry-Esseen误差界。指数逼近及其在Bernoulli-Laplace马尔可夫链谱中的应用也进行了讨论。
英文摘要：
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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