Suppose we have $X_1,X_2,...$ Bernoulli RV (not necessarily independent) with $P(X_i=1)=p_i$. I am trying to understand what conditions are sufficient in order for a CLT to hold.
Surely if the sequence of variances does not converge to $C>0$ a CLT will not hold, but suppose that $\frac{1}{\sqrt{n}}Var(\Sigma_{i=1}^n X_i)\overset{n\longrightarrow \infty}{\longrightarrow}p$, does a CLT hold in this case?