Let $\alpha \neq 1.$


**If $X,Y$ are two independent random variable such that $U=X+Y$ and $V=X+\alpha Y$ are independent, then $X$ and $Y$ are normally distributed.**


In term of characteristic functions this means that $$\forall x, y \in \mathbb{R}, \phi\_X(x+y)\phi\_Y(x+\alpha y)=\phi\_X(x)\phi\_Y(x)\phi\_X(y)\phi\_Y(\alpha y) \ \ \ \ \ \ \ \ (E)$$


The particular case where $\alpha=-1$ was treated [here](https://math.stackexchange.com/questions/3822099/normal-distribution-magic).


Trying to use that way for the general case didn't work.


**The main problem is how to solve the functional equation $(E).$**


Is there any known way to solve the equation analytically? Is it possible to find a functional equation $(E\_1)$ which only depends of $\phi\_X$ and $(E\_2)$ which only depends of $\phi\_Y$?


(We should find $|\phi\_X(x)|=e^{x^2c\_1},|\phi\_Y(x)|=e^{c\_2x^2},c\_1,c\_2 \leq 0$)



Functional equations and normal distribution
Let $a:=\alpha$.


If $a\ne0$, then $X$ and $Y$ are normal by the [Darmois--Skitovich theorem](https://en.wikipedia.org/wiki/Darmois%E2%80%93Skitovich_theorem).


If $a=0$ and the distribution of $X$ is nondegenerate, then $U=X+Y$ and $V=X$ cannot be independent.


If $a=0$ and the distribution of $X$ is degenerate, then $U=X+Y$ and $V=X$ are independent for any random variable $Y$.


