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