Semi-G-normal: a Hybrid between Normal and G-normal
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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.
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