Semi-$G$-normal: a Hybrid between Normal and $G$-normal (Full Version)

TL;DR

This paper introduces semi-$G$-normal distributions, a hybrid between classical normal and $G$-normal, to better understand distributional uncertainty and independence in the $G$-expectation framework.

Contribution

It proposes semi-$G$-normal as a new distributional structure with variance uncertainty but zero skewness, bridging classical normal and $G$-normal distributions.

Findings

Semi-$G$-normal has variance uncertainty but zero skewness.

Representations of semi-$G$-normal vectors under various independence types.

Connections to state-space volatility models with a common graphical structure.

Abstract

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.