$X$ is s.t. $\mathbb{E}(X) = 0, \mathbb{E}(X^{2}) < \infty$. $X, Y$ are i.i.d.
If $(\frac{X+Y}{\sqrt{2}}, \frac{X-Y}{\sqrt{2}})$ has the same distribution as $(X, Y)$, then $X$ is a normal r.v.
Where should I start? Appreciate any hint!
5
-
$\begingroup$ Your first bet should be to deduce as many facts about the characteristic function of $X$ as possible. $\endgroup$– AndrewNov 27, 2022 at 0:44
-
$\begingroup$ In particular, what is $\varphi_X(2u)?$ $\endgroup$– AndrewNov 27, 2022 at 0:50
-
$\begingroup$ @AndrewZhang I think by a theorem I can write $\phi(t) = 1 - \frac{1}{2}t^{2}\mathbb{E}(X^{2}) + o(t)t^{2}$ $\endgroup$– TomNov 27, 2022 at 1:25
-
1$\begingroup$ $(X+Y)/\sqrt 2$ has the same characteristic function as $X$, and so does $(X-Y)/\sqrt 2$, which is also independent. So... $\endgroup$– AndrewNov 27, 2022 at 3:06
-
$\begingroup$ @AndrewZhang Ah I remember something. Is it CLT? $\endgroup$– TomNov 28, 2022 at 3:29
Add a comment
|