A Note On $G$-normal Distributions

TL;DR

This paper investigates the conditions under which the convolution of two $G$-normal distributions results in a $G$-normal distribution, revealing a specific ratio condition for their variance intervals.

Contribution

It establishes a precise necessary and sufficient condition for the convolution of two $G$-normal distributions to remain $G$-normal.

Findings

Convolution of $G$-normal distributions may not be $G$-normal with different variance intervals.

The convolution is $G$-normal if and only if the ratio of upper to lower variance bounds is equal for both distributions.

Provides a clear criterion for the stability of $G$-normality under convolution.

Abstract

As is known, the convolution $\mu*\nu$ of two $G$-normal distributions $\mu, \nu$ with different intervals of variances may not be $G$-normal. We shows that $\mu*\nu$ is a $G$-normal distribution if and only if $\frac{\overline{\sigma}_\mu}{\underline{\sigma}_\mu}=\frac{\overline{\sigma}_\nu}{\underline{\sigma}_\nu}$.