Let $U$ denote a random variable with PDF $$f_U(u)=ce^{-u\sqrt{u}}\,\mathbf 1_{u>0}$$ Does there exist a random variable $W$ independent of $U$ such that $U+W$ is distributed like $2U$?
This question is related to something I asked here.
In other words, can we find a random variable $W$ such that \begin{align} 2 U' =U+W \end{align} where $W$ and $U$ are independent and $U'$ has the same distribution as $U$.
In general the answer is such a $W$ doesn't exist.
To see this let assume $U_1$ and $U_2$ are independent and $W$ exists. Then $$U_2 = 2U_1 - W$$ but $2U_1 - W$ is independent of $U_2$ by assumption so $U_2$ is independent of itself hence $U_2$ is constant what is a contradiction.
So such a $W$ cannot exist.