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 $\widehat{\mathbf{E}}$ with $\widehat{\mathbf{E}}[X_1]=\widehat{\mathbf{E}}[-X_1]=0$ and $\widehat{\mathbf{E}}[\mathrm{\mid }{X}_1{\mathrm{\mid }}^3]<\mathrm{\infty }$. Setting $W_n:=\frac{X_1+\cdots +X_n}{\sqrt{n}}$, we have, for each bounded and Lipschitz function $\phi$, $$\underset{n\to \mathrm{\infty }}{\lim }\mid \widehat{\mathbf{E}}[\phi (W_n)]-{\mathcal{N}}_G(\phi )\mid =0,$$ where ${\mathcal{N}}_G$ is the $G$-normal distribution with $G(a)=\frac{1}{2}\widehat{\mathbf{E}}[a{X}_1^2]$, $a\in \mathbb{R}$.} 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 $\underset{‾}{\sigma }$ and $\bar{\sigma }$, and a positive constant $C_{\alpha ,G}$ depending on $\alpha ,\underset{‾}{\sigma }$ and $\bar{\sigma }$ such that $$\underset{\mathrm{\mid }\phi {\mathrm{\mid }}_{Lip}\le 1}{\sup }\mid \widehat{\mathbf{E}}[\phi (W_n)]-{\mathcal{N}}_G(\phi )\mid \le C_{\alpha ,G}\frac{\widehat{\mathbf{E}}[\mathrm{\mid }{X}_1{\mathrm{\mid }}^{2+\alpha }]}{n^{\frac{\alpha }{2}}},$$ where ${\bar{\sigma }}^2=\widehat{\mathbf{E}}[X_1^2]$, ${\underset{‾}{\sigma }}^2=-\widehat{\mathbf{E}}[-X_1^2]>0$ and ${\mathcal{N}}_G$ is the $G$-normal distribution with $$G(a)=\frac{1}{2}\widehat{\mathbf{E}}[a{X}_1^2]=\frac{1}{2}({\bar{\sigma }}^2{a}^{+}-{\underset{‾}{\sigma }}^2{a}^{-}),a\in \mathbb{R}\mathrm{.}$$}