## Electronic Communications in Probability

### 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 $\mu ^\varphi [\varphi ]=\mathcal{N} [\varphi ]$.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 24, 12 pp.

Dates
Accepted: 4 April 2017
First available in Project Euclid: 19 April 2017

https://projecteuclid.org/euclid.ecp/1492588926

Digital Object Identifier
doi:10.1214/17-ECP53

Mathematical Reviews number (MathSciNet)
MR3645506

Zentralblatt MATH identifier
1370.60020

#### Citation

Hu, Mingshang; Peng, Shige; Song, Yongsheng. Stein type characterization for $G$-normal distributions. Electron. Commun. Probab. 22 (2017), paper no. 24, 12 pp. doi:10.1214/17-ECP53. https://projecteuclid.org/euclid.ecp/1492588926

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