Sum of a Gumbel and Logistic random variable It is known that the difference between two Gumbel random variables with the same variance is a Logistic with the same variance as the Gumbel random variables.
Is it true that the sum of a Gumbel and a Logistic random variables, with the same variance, is a Gumbel?
 A: For the sake of simplicity, let $X_1, X_2 \sim \operatorname{Gumbel}(0,1)$ be iid Gumbel random variables with location parameter $0$ and scale parameter $1$; i.e., $$F_X(x) = \exp\left(-e^{-x}\right), \quad x \in \mathbb R.$$  Let $Y_i = e^{-X_i}$, so that $$F_Y(y) = \Pr[Y \le y] = \Pr[X \le -\log y] = F_X(-\log y) = e^{-y}, \quad y > 0.$$  Thus $Y_1, Y_2 \sim \operatorname{Exponential}(1)$.  If $W = Y_1/Y_2$, we then compute $$F_W(w) = \Pr[W \le w] = \Pr[Y_1/Y_2 \le w] = \int_{y_1 = 0}^\infty \int_{y_2 = y_1/w}^\infty e^{-y_1} e^{-y_2} \, dy_2 \, dy_1 = \frac{w}{1+w}, \quad w > 0.$$  Therefore, $$Z = X_2 - X_1 = \log(e^{-X_1}/e^{-X_2}) = \log \frac{Y_1}{Y_2}$$ has CDF $$F_Z(z) = \Pr[Z \le z] = \Pr[W \le e^z] = \frac{e^z}{1+e^z} = \frac{1}{1+e^{-z}}, \quad z \in \mathbb R,$$ hence $$Z = X_2 - X_1 \sim \operatorname{Logistic}(0,1),$$ as claimed.  This proof generalizes without much difficulty to arbitrary location parameters on the $X_i$.
Let us consider the reverse:  with the variables defined as above, we are interested in the distribution of $$G = X + Z = \log(e^{X + Z}) = \log(W/Y),$$ so let $H = e^G = W/Y$ and compute $$F_H(h) = \Pr[W/Y \le h] =  \int_{w=0}^\infty \int_{y=w/h}^\infty f_Y(y) f_W(w) \, dy \, dw \\
= \int_{w=0}^\infty \frac{e^{-w/h}}{(1+w)^2} \, dw, \quad h > 0.$$  Then $G$ has CDF $$F_G(g) = \Pr[G \le g] = \Pr[H \le e^g] = F_H(e^g) = \int_{w=0}^\infty \frac{e^{-w/e^g}}{(1+w)^2} \, dw,$$ but it should be clear that this has no elementary closed form solution; thus we have no hope that $G$ is Gumbel distributed.
