We know that $X\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ and $Y\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ then $X+Y\sim {\mathrm {Logistic}}(2\alpha ,\beta )$.
I am wondering, what will be $X+Y+Z$ like, if $Z\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ also. And more generally what will $\sum X_{i}$ be if $X_{i}\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$.
A practical example would be a device whose lifespan follows $X\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$. But the device can be repaired, instantaneously once failed, for $i$ times.
