Suppose we have a random variable $W$ and we want to transform it to a random variable $V$ by using additive transformation, as follows \begin{align} V=U+W \end{align} where $U$ is independent of $W$.
My question is: We are interested in knowing whether such random variable $U$ exists or not and what methods can we used to check this.
Known method: For me, the most obvious way is via characteristic functions.
\begin{align} \phi_V(t)=\phi_U(t) \cdot \phi_W(t) \to \phi_U(t)=\frac{\phi_V(t)}{\phi_W(t)} \end{align} and we have to check if $\phi_U(t)$ is a proper characteristic function.
Motivation: In order to show that there is no such transformation all we need is to check that $\phi_U$ violates one of the properties of characteristic function. For example, by showing that $|\phi_U(t)|>1$ for some $t$.
To show that such a transformation indeed exists one has to check that $\phi_U$ is a proper charcterstic function, which can be done via the following set of theorems (see here).
Unfortunately, these theorems can be difficult to check and I was wondering if there is another method (not through characteristic functions) that can be used to prove or disprove existence of such transformations.
For me, the more important direction is how to disprove the existence, which I guess amounts to showing some necessary condition and then checking if it holds.
Proposed Example
I know that the question I am asking can be fairly difficult to answer. So it might be a good idea to focus on a specific example:
\begin{align} 2V=U+W \end{align} where \begin{align} V &\sim c_ve^{-v^{1.5}}, v\ge0\\ W & \sim c_We^{-w^{1.5-\epsilon_W}}, w\ge0 \end{align} for some $\epsilon_W \in (-\infty, 1.5)$ and \begin{align} c_W= \frac{1.5-\epsilon_W}{\Gamma \left(\frac{1}{1.5-\epsilon_W}\right)} \end{align}
Why did I pick this example: On the one hand, computing anaylytic expression for the ChF's of this example might be impossible and motivates searching for new methods. On the other hand, becuse pdf has an exponential which usually has some nice properties, make me think that there might be hope of answering this question.
To solve the above question we distinguess three case: $\epsilon_W <0$, $\epsilon_W =0$, $1.5\ge \epsilon_W >0$.
Solution for $1.5\ge \epsilon_W >0$. Via a method outlined in one of the anwer we have that for any $n$ \begin{align} 2 \ge \left(\frac{E[V^n]}{E[W^n]} \right)^{1/n}. \end{align} For $1.5\ge \epsilon_W >0$ we can show that the RHS of the above equation can be made as larege as possible by choosing appropriate $n$ and we reach a contradiction.
So, for the case of $1.5\ge \epsilon_W >0$ transformation $2V=U+W$ is impossible.
At this point, however, I do not know how to apporach the case of $\epsilon_W <0$ and the case of $\epsilon_W =0$.
To avoid a really long question. The case of $\epsilon_W =0$ was asked here.
