For i.i.d X and Y , if X + Y and X - Y are independent, show X is normally distributed [closed]

Question

The question goes as follows: If $X$ and $Y$ are independent and identically distributed, their density function $f(x)$ is strictly positive and second-order continuously differentiable. If $X+Y$ and $X-Y$ are independent, show that $X$ is normally distributed.

Given the conditions above, I can get the equation that the characteristic function of $X$ satisfies: $$ \phi(2t) = \phi^3(t)\phi(-t) $$ But I don't know how to recover the specific form of this function simply from this equation.

Could anybody give me a hand? Thanks a lot!

Answer 1

This is a very special case of the Darmois-Skitovich theorem, which deals with two arbitrary linear forms of the independent random variables, without any a priori conditions about their distributions. See, for example

Kagan, A.M., Linnik, Yu.V.; Rao, C.Radhakrishna Characterization problems in mathematical statistics,

or perhaps more accessible

Theory Probab. Appl., 57(3), 368–374. I. A. Ibragimov, On the Skitovich--Darmois--Ramachandran Theorem Theory Probab. Appl., 57(3), 368–374.

Or just type "Darmois-Skitovich theorem" on Google.