The $G$-expectation framework is a generalization of the classical probabilistic system motivated by Knightian uncertainty, where the $G$-normal plays a central role. However, from a statistical perspective, $G$-normal distributions look quite different from the classical normal ones. For instance, its uncertainty is characterized by a set of distributions which covers not only classical normal with different variances, but additional distributions typically having non-zero skewness. The $G$-moments of $G$-normals are defined by a class of fully nonlinear PDEs called $G$-heat equations. To understand $G$-normal in a probabilistic and stochastic way that is more friendly to statisticians and practitioners, we introduce a substructure called semi-$G$-normal, which behaves like a hybrid between normal and $G$-normal: it has variance uncertainty but zero-skewness. We will show that the non-zero skewness arises when we impose the $G$-version sequential independence on the semi-$G$-normal. More importantly, we provide a series of representations of random vectors with semi-$G$-normal marginals under various types of independence. Each of these representations under a typical order of independence is closely related to a class of state-space volatility models with a common graphical structure. In short, semi-$G$-normal gives a (conceptual) transition from classical normal to $G$-normal, allowing us a better understanding of the distributional uncertainty of $G$-normal and the sequential independence.

翻译：$G$的预期框架是典型的概率体系的笼统化,其动机是骑士不确定,通常的美元在其中扮演着核心角色。然而,从统计角度看,正常的美元分配看起来与典型的正常分配大不相同。例如,其不确定性的特点是一系列分配结构,不仅包括传统常态,有不同差异,而且包括额外分配,通常不是零偏差。美元正常的美元趋势由完全非线性PDE的类别“G$-热方程”来定义。为了理解对统计人员和从业者来说更为友好的概率性和随机性分配,我们采用了一套称为半G美元正常的次级结构,这种结构不仅在正常和美元正常之间混合,而且具有非零偏差性。我们将显示,当我们在半G$的正常度上将G$的顺序独立置于一个完全非直线性类别。 更为重要的是,我们提供了一种相对正常的G美元结构的正统性结构下,一种相对的正统性结构,即从普通的正值结构的正值结构的每类中,一种相对的正统的正值结构的正向。