Let random variable X1, with variance one, has the following property: $\frac {X1+X2} {\sqrt2}$ has the same distribution as X1, where X2 is an independent copy of X1. Show that X1 ∼ N(0, 1).
So i have to show x1 is normal distribution; its very easy to show the other way of course, when x1 is normal distirbution, then obviously $\frac {X1+X2} {\sqrt2}$ ~ X1. But the other way is much harder
DOes this have to with characteristic function? since $\phi_{x1+x2}=\phi_x\phi_ {x2}$ i had written out that expression but how can i transofrm it to normal distirbution? in part, how do i realize the pdf of normal distribution from the equality here $\phi_{x1}=\phi_{x1/\sqrt(2)}\phi_ {x2\sqrt(2)}$? thanks in advance.
Let $X_1,X_2,X_3,\dots$ be an i.i.d. series where each variable is distributed like $X_1$. By iterating the equation $X_1\stackrel{d}=(X_1+X_2)/\sqrt{2}$, we get that for any $n\in \mathbb N$,
$$ X_1\stackrel{d}=\frac{X_1+X_2+\dots+X_{2^n}}{\sqrt{2^n}} $$ The central limit theorem implies that the RHS of the above approaches an $N(0,1)$ distribution as $n\to\infty$, so we conclude that $X_1\sim N(0,1)$. In order to apply CLT, we only need $X_1$ to have mean zero (easy to prove) and variance one (given).
The comments below this question are now irrelevant, since I replaced my earlier answer with this new one. It turns out my earlier answer was mistaken; it assumed that all higher moments of $X_1$ existed without proof.