A note on characterizations of G-normal distribution

TL;DR

This paper characterizes G-normal distributions by examining how the distribution of a transformed variable remains invariant with respect to a parameter, extending previous results without requiring identical distribution of variables.

Contribution

It introduces new characterizations of G-normality based on distribution invariance under certain transformations, even when variables are not identically distributed.

Findings

Characterization of G-normality via distribution invariance.

Extension of results without the identical distribution assumption.

Provides a new perspective on G-normal distribution properties.

Abstract

In this paper, we show that the G-normality of X and Y can be characterized according to the form of f such that the distribution of {\lambda}+f({\lambda})Y does not depend on {\lambda}, where Y is an independent copy of X and {\lambda} is in the domain of f. Without the condition that Y is identically distributed with X, we still have a similar argument.