A Note On G-normal Distributions

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 $\frac{\overline{\sigma}_\mu}{\underline{\sigma}_\mu}=\frac{\overline{\sigma}_\nu}{\underline{\sigma}_\nu}$.