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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.

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 $|\phi_X(x)|=e^{x^2c_1},|\phi_Y(x)|=e^{c_2x^2},c_1,c_2 \leq 0$)

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Let $a:=\alpha$.

If $a\ne0$, then $X$ and $Y$ are normal by the Darmois--Skitovich 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 $Y$.

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  • $\begingroup$ there are several ways to prove Darmois-Skitovich theorem, one of them is by taking in $(E)$ the $\log$ to prove that we have a bilinear form, another way is the use of Cramer theorem (complex analysis), so I was wandering if there are other analytical ways which include solving functional equations $\endgroup$
    – Kurt.W.X
    Commented Mar 7, 2021 at 22:22
  • $\begingroup$ @Kurt.W.X : I don't understand your comment. In your post, Darmois-Skitovich theorem was not even mentioned. Neither was the word "other" or "another" used in your post. Instead, you said "The main problem is how to solve the functional equation $(E)$." The more general functional equation than $(E)$ is solved e.g. in a the proof of the Darmois-Skitovich theorem in the book by Kagan, Linnik, and Rao -- see equation (3.1.5) there. $\endgroup$
    – Iosif Pinelis
    Commented Mar 7, 2021 at 22:50
  • $\begingroup$ Thanks for the reference ! $\endgroup$
    – Kurt.W.X
    Commented Mar 8, 2021 at 16:38
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The general reference for this type of theorems, and related functional equations is A. Kagan, Yu. Linnik and C. Rao, Characterization problems of mathematical statistics, K. Wiley and Sons, NY, 1973 (translated from the Russian). The result about independence of two linear forms is called the Darmois-Skitowich Theorem.

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