Comparing the $G$-Normal Distribution to its Classical Counterpart

TL;DR

This paper explores the properties of the multidimensional $G$-normal distribution, revealing nonintuitive behaviors and extending classical results to a nonlinear, uncertain covariance setting.

Contribution

It advances understanding of the multidimensional $G$-normal distribution by addressing independence, covariance uncertainty, and linear transformations, highlighting novel nonintuitive properties.

Findings

Multidimensional $G$-normal exhibits unexpected behaviors.

Classically-inspired properties extend to the nonlinear $G$-framework.

New insights into independence and covariance uncertainty in $G$-normal.

Abstract

In one dimension, the theory of the $G$-normal distribution is well-developed, and many results from the classical setting have a nonlinear counterpart. Significant challenges remain in multiple dimensions, and some of what has already been discovered is quite nonintuitive. By answering several classically-inspired questions concerning independence, covariance uncertainty, and behavior under certain linear operations, we continue to highlight the fascinating range of unexpected attributes of the multidimensional $G$-normal distribution.