Moments of G-normal distributions with respect to Choquet expectations

In this paper, we examine two types of nonlinear expectations: Choquet expectation and sublinear expectation. Similar to the role played by the normal distribution and Wiener space in classical probability theory, the $G{\text{-normal}}$ distribution and $G{\text{-expectation}}$ space are crucial in the theory of sublinear expectation. Our objective is to investigate the relationship between moments of variables following a $G{\text{-normal}}$ distribution under Choquet expectation and sublinear expectation. We provide bounds for the $p{\text{-order}}$ absolute moment (where $p>0$) of a $G{\text{-normal}}$ distribution, specifically under Choquet expectation. Notably, we derive a formula for the Choquet expectation of $\varphi(X)$, where $\varphi$ is a strictly monotonic function and $X$ is a $G{\text{-normally}}$ distributed random variable. As a special case, we obtain the explicit expressions for the odd moments of a $G{\text{-normal}}$ distribution corresponding to Choquet expectation. Consequently, we establish that if the $G{\text{-expectation}}$ coincides with a Choquet expectation, then the $G{\text{-expectation}}$ is linear.