Showing that two unknown random variables are both Gaussian, if their sum is Gaussian

Question

I'm currently studying probability theory and came across this exercise question that I'm struggling to find the answer to.

I've found many resources regarding showing that the sum of two normally distributed random variables is also normally distributed, but am struggling to find anything that goes the opposite way.

The specific problem is as follows:

Let $Y = X_1 + X_2$, where $X_1$ and $X_2$ are unknown i.i.d. random variables. Show that $X_1$ and $X_2$ are Gaussian if $Y$ is Gaussian.

I'm fairly lost and am having trouble on where to start. Any tips or advice are appreciated. Thanks in advance.

Answer 1

If you are able to use characteristic functions, note that i.i.d.-ness of $X_1$ and $X_2$ implies $$E[e^{itY}] = E[e^{itX_1} e^{itX_2}] = E[e^{itX_1}] E[e^{itX_1}] = E[e^{itX_1}]^2,$$ and then use the fact that $Y$ is Gaussian to conclude that $X_1$ is Gaussian.