CLT for the squares: a finite fourth moment gives a normal limit

Let $\{X_n,n\ge1\}$ be a sequence of independent identically distributed random variables with finite (non-zero) fourth moment. Let $Y_n=(X_1^2+\dots+X_n^2)/n$. Which of the following is true? As $n\to\infty$, $\sqrt n\bigl(Y_n-E[X_1^2]\bigr)$: