A Note On $G$-normal Distributions

As is known, the convolution $\mu \ast \nu$ of two $G$-normal distributions $\mu, \nu$withdifferentintervalsofvariancesmaynotbe$G$-normal\mathrm{.}Weshowsthat$\mu*\nu$isa$G$-normaldistributionifandonlyif$\frac{\overline{\sigma}\mu}{\underline{\sigma}\mu}=\frac{\overline{\sigma}\nu}{\underline{\sigma}\nu}$.