Stein Type Characterization for $G$-normal Distributions

TL;DR

This paper establishes a Stein type characterization for G-normal distributions within sublinear expectation spaces, providing a new criterion involving integral equations that uniquely identify G-normality.

Contribution

It introduces a novel Stein type characterization for G-normal distributions, linking sublinear expectations with integral conditions involving test functions.

Findings

Provides a necessary and sufficient condition for G-normality

Connects G-normal distributions with integral equations involving test functions

Enhances understanding of G-normal distributions in sublinear expectation theory

Abstract

In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N}[\varphi]=\max_{\mu\in\Theta}\mu[\varphi],\ \varphi\in C_{b,Lip}(\mathbb{R}),$ 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 $\mu^\varphi[\varphi]=\mathcal{N}[\varphi]$.