Suppose $X_1,X_2,...$ are Bernoulli random variables with $P(X_i=1)=p_i$ and $X_i$ have negative correlation. Is there a CLT in this case, i.e. does $\frac{Z_n-(\Sigma^n_{i=1}p_i)}{\sqrt{n}}$ converge to a Gaussian in distribution?
And if we add the assumption that $\underset{n \longrightarrow \infty}{lim} \frac{\Sigma^n_{i=1}p_i}{n}=p\in(0,1)$, is it true then?
No, the CLT need not hold under these assumptions. Consider the following example: take $p=1/2$ for definiteness, and divide the (discrete) time into intervals $I_1=[1,2]$, $I_n=(2^{n-1}, 2^n]$, $n\geq 2$. On each interval, let $(X_k, k\in I_n)$ be the i.i.d. Bernoulli($1/2$) conditioned on $\sum_{k\in I_n}X_k=|I_n|/2$ (think about placing $|I_n|/2$ balls into $|I_n|$ urns at random); clearly, $(X_i,X_j)$ are negatively correlated when $i,j\in I_n$ for some $n$. Then the CLT doesn't hold since the sum becomes deterministic from time to time.
The correlations are not strictly negative though (because the $X$'s are independent if belong to different intervals), but you can probably make them so by some "small" perturbation.
