Normal Approximation by Stein's Method under Sublinear Expectations

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 $$G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overline{\sigma}^2a^+-\underline{\sigma}^2a^-), \ a\in \mathbb{R}.$$}