CLT infinite mean

Question

I am trying to see if we can construct $(X_n)_{n\ge 1}$, a sequence of i.i.d random variables such that $E|X_n|=\infty$ while $n^{-1/2}(X_1+...+X_n)$ converges in distribution goes to a $N(0,1)$ random variable. I tried using $$ n^{1/2}(1/n(X_1+..+X_n))$$ and using SLLN on the inside but then the limit would vanish. Is it even possible to construct such a sequence?

Answer 1

No, it is not possible. For the CLT, we need to have almost two finite absolute moments and if the first absolute moment is infinite, so are all higher order moments (this follows from Hölder's inequality). Hence, the CLT does not hold.