If $Z_1$ and $Z_2$ are two $L^1$ real-values random variables, it is well known that
$ES_\alpha(Z_1+Z_2) \le ES_\alpha(Z_1) + ES_\alpha(Z_2)$ for any $\alpha \in (0,1)$
However, what about the equality case? I see two cases:
- one of the random variables is constant
- random variables are proportional
Are there other cases?