So this is a problem I saw in the AoPs forum. Though they didn't solve it. My teacher assistant found a proof, which I found nice !
Here is the problem.
is a sequence of iid random variables, following a standard normal distribution
Find the characteristic function of the couple
As an extra, you can also find the joint pdf of the couple, by using the independence between and
And you can also try to prove that independence (that's a consequence of Cochran's theorem, but you can do it without it)