I would like your help to derive the distribution of the sum/difference of two random variables which are distributed as Gumbel.
Consider $X$ having Gumbel distribution with "long tail on the right" as defined here, location $\gamma$ (Euler constant), and scale $\beta_x>0$.
Consider $Y$ having Gumbel distribution with "long tail on the right" as defined here, location $\gamma$ (Euler constant), and scale $\beta_y>0$.
(1) What is the distribution of $X-Y$?
According to Wikipedia, if $\beta_x=\beta_y=\beta$, then $X-Y$ is Logistic with location $\gamma-\gamma=0$ and scale $\beta$.
Can we say something when $\beta_x\neq \beta_y$?
(2) What is the distribution of $X+Y$? My thoughts:
(2.1) $X+Y=X-(-Y)$
(2.2) $-Y$ is Gumbel with location $\gamma$ and scale $-\beta_y$
(2.3) Hence ?