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?



CLT for Bernoulli RV with negative correlation
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.


