Peng (2008)(\cite{P08b}) proved the Central Limit Theorem under a sublinear expectation: \textit{Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d random variables under a sublinear expectation $\hat{\mathbf{E}}$ with $\hat{\mathbf{E}}[X_1]=\hat{\mathbf{E}}[-X_1]=0$ and $\hat{\mathbf{E}}[|X_1|^3]<\infty$. Setting $W_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$, we have, for each bounded and Lipschitz function $\varphi$, \[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|=0,\] where $\mathcal{N}_G$ is the $G$-normal distribution with $G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]$, $a\in \mathbb{R}$.} In this paper, we shall give an estimate of the rate of convergence of this CLT by Stein's method under sublinear expectations: \textit{Under the same conditions as above, there exists $\alpha\in(0,1)$ depending on $\underline{\sigma}$ and $\overline{\sigma}$, and a positive constant $C_{\alpha, G}$ depending on $\alpha, \underline{\sigma}$ and $\overline{\sigma}$ such that \[\sup_{|\varphi|_{Lip}\le1}\bigg|\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|\leq C_{\alpha,G}\frac{\hat{\mathbf{E}}[|X_1|^{2+\alpha}]}{n^{\frac{\alpha}{2}}},\] where $\overline{\sigma}^2=\hat{\mathbf{E}}[X_1^2]$, $\underline{\sigma}^2=-\hat{\mathbf{E}}[-X_1^2]>0$ and $\mathcal{N}_G$ is the $G$-normal distribution with \[G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overline{\sigma}^2a^+-\underline{\sigma}^2a^-), \ a\in \mathbb{R}.\]}

中文翻译：

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次线性期望下 Stein 方法的正态逼近

Peng (2008)(\cite{P08b}) 证明了亚线性期望下的中心极限定理：\textit{让 $(X_i)_{i\ge 1}$ 是一个亚线性期望下的 iid 随机变量序列 $\ hat{\mathbf{E}}$ 与 $\hat{\mathbf{E}}[X_1]=\hat{\mathbf{E}}[-X_1]=0$ 和 $\hat{\mathbf{E} }[|X_1|^3]<\infty$。设置 $W_n:=\frac{X_1+\cdots+X_n}{\sqrt{n}}$，对于每个有界和 Lipschitz 函数 $\varphi$，\[\lim_{n\rightarrow\infty}\bigg |\hat{\mathbf{E}}[\varphi(W_n)]-\mathcal{N}_G(\varphi)\bigg|=0,\] 其中 $\mathcal{N}_G$ 是 $G$ -正态分布与 $G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]$, $a\in \mathbb{R}$.} 在本文中，我们将在次线性期望下通过 Stein 方法给出对这个 CLT 收敛率的估计： \textit{在与上述相同的条件下，存在 $\alpha\in(0,