Stein type characterization for $G$-normal distributions

Abstract

In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N} [\varphi ]=\sup _{\mu \in \Theta }\mu [\varphi ],\ \varphi \in C_{b,Lip}(\mathbb{R} ),$ be a sublinear expectation. $\mathcal{N} $ is $G$-normal if and only if for any $\varphi \in C_b^2(\mathbb{R} )$, we have \[ \int _\mathbb{R} [\frac{x} {2}\varphi '(x)-G(\varphi ''(x))]\mu ^\varphi (dx)=0, \] where $\mu ^\varphi $ is a realization of $\varphi $ associated with $\mathcal{N} $, i.e., $\mu ^\varphi \in \Theta $ and $\mu ^\varphi [\varphi ]=\mathcal{N} [\varphi ]$.