This is an old qualifying exam problem of probability theory.
Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,\frac{X+Y}{\sqrt{2}}$ are all identically distributed. Prove that $X\sim N(0,1)$.
My attempt: Let $\varphi$ be the characteristic function of $X$. Then by the given condition, I got $\varphi(t)=\varphi(t/\sqrt{2})^2$. Then I'm trying to conclude $\varphi(t)=e^{-t^2/2}$ from such functional equation, but it seems not work.
Does anyone have ideas?
Thanks in advance!
Iterating the property of $\varphi$ found in the question, one gets $\varphi(t)=\left(\varphi(t/2^{n/2})\right)^{2^n}$ for all $n\in\mathbb{N}$.
Also, since $X$ has mean zero and variance 1, $\varphi(t)=1-\frac{t^2}{2}+o(t^2)$ as $t\to 0$.
Therefore, $$\varphi(t)=\left(1-\frac{t^2}{2^{n+1}}+o(\frac{t^2}{2^n})\right)^{2^n}=\lim_{n\to\infty}\left(1-\frac{t^2}{2^{n+1}}+o(\frac{t^2}{2^n})\right)^{2^n}=e^{-t^2/2} $$ as desired.