Hi I am looking for a result (if it exists !!!) in the direction of Normal approximation for sum of correlated Bernoulli random variables (edit : with the same parameter $p$) where correlation between any pair of distinct Bernoulli variables entering into the summand is constant and equal to $0<\rho<1$.
I have googled the question but with no success, if any has reference or result in this particular case I would be grateful.
Edit as suggested by Glen_b here is my results for mean and variance calculations (so correlation increases the variance) :
$$E[\sum_{i=1}^{n}X_i]=n.p$$
$$E[(\sum_{i=1}^{n}X_i)^2]-E[\sum_{i=1}^{n}X_i]^2=n.p.(1-p)+ n.(n-1)\rho.p.(1-p)$$
$$=np.(1-p)(1+\rho(n-1))$$
Best regards
Best Answer
If the number of variables is sufficiently large and the correlation is bounded away from 1, then there are Central Limit Theorems that apply (e.g., also see versions of the CLT for stationary processes).
So if your $i$-th variable has parameter $p_i$, then the variance of $X_i$ is $p_i(1-p_i)$ and :
$$\text{Cov}(X_i,X_j)=\rho \sqrt{p_i(1-p_i)\cdot p_j(1-p_j)}$$
The expected value of the sum is the sum of the expected values. The variance of the sum is the sum of the variances plus twice the sum of all the pairwise covariances, and if $n$ is large enough, the standardized sum will be approximately standard normal.
You may want to consider the possibility of using a continuity correction.