Normal Approximation by Stein's Method under Sublinear Expectations

Abstract

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 $\phi$, \[\lim_{n\rightarrow\infty}\bigg|\hat{\mathbf{E}}[φ(W_n)]-\mathcal{N}_G(φ)\bigg|=0,\] where ${\mathcal{N}}_G$ is the $G$-normal distribution with $G(a)=\frac{1}{2}\hat{\mathbf{E}}[a{X}_1^2]$, $a\in \mathbb{R}$.} 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 $\underline{\sigma }$ and $\overline{\sigma }$, and a positive constant $C_{\alpha ,G}$ depending on $\alpha ,\underline{\sigma }$ and $\overline{\sigma }$ such that \[\sup_{|φ|_{Lip}\le1}\bigg|\hat{\mathbf{E}}[φ(W_n)]-\mathcal{N}_G(φ)\bigg|\leq C_{α,G}\frac{\hat{\mathbf{E}}[|X_1|^{2+α}]}{n^{\fracα{2}}},\] 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 \[G(a)=\frac{1}{2}\hat{\mathbf{E}}[aX_1^2]=\frac{1}{2}(\overlineσ^2a^+-\underlineσ^2a^-), \ a\in \mathbb{R}.\]}

References in corpus (3)

• A New Central Limit Theorem under Sublinear Expectations
• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
• Dynamically Consistent Nonlinear Evaluations and Expectations

Cited by in corpus (1)

• Stein's Method for Law of Large Numbers under Sublinear Expectations