Recall that the classical Central Limit Theorem states that for any sequence of I.I.D random variables $ X_{i} \sim X $ such that $E(X)=0,Var(X)=1$ we have $\frac{\sum_{i=1}^{n}X_{i}}{\sqrt{n}} \rightarrow^{d} N(0,1)$. Suppose I have a sequence of random variables that are only known to be uncorrelated and centered (i.e, $E(X_{i})=0$, and $E(X_{i}X_{j})=0$ for $i \neq j$). One can also assume (if this helps) that the higher moments of the variables are finite. I know that the CLT itself does not apply in this case, but can one still get something weaker, namely that $$\frac{\sum_{i=1}^{n}X_{i}}{\sqrt{n}} \rightarrow^{d} Y$$
for some random variable Y?