I am struggling to find a suitable Central Limit Theorem (CLT) for dependent variables for the following example:
We have $K$ bins. In each bin $B_i$, $i=1,...,K$, we have $N_i$ balls, with $N=\sum_{i=1}^KN_i$. We want to sample $n$ balls in total. Fix bin $B_i$, then we have the random variables $I(X_1 \in B_i),..., I(X_n \in B_i)$, where they each represent a Bernoulli distribution with probability $p_i=N_i/N$ of taking a ball from bin $B_i$. Even though they are identically distributed, they are not independent. This is essentially a multivariate hypergeometric setting.
The goal is to find a function of the sum of these dependent Bernoulli random variables that goes to a normal distribution in its limit. (Similar to the proof of the Chi-squared goodness-of-fit test, but now with dependent Bernoulli random variables).
I found several versions of the CLT for dependent random variables, but they either require zero expectation, m-dependence, or pairwise independence. But I haven't found anything yet that is suitable for this example.