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There are two sets of random variables $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ satisfying:

  1. Each $X_i$ and each $Y_j$ has a symmetric Bernoulli distribution ($-1$ and $+1$ with probability $\frac12$ each).
  2. $X_1,\ldots,X_n$ are independent. Similarly, $Y_1,\ldots,Y_n$ are independent.
  3. For each $i,j$, $cov(X_i,Y_j) = c/n$ for some small constant $c\gt 0$.

Now define $X=\sum_i X_i$ and $Y=\sum_j Y_j$. Each of $X$ and $Y$ has a shifted and scaled binomial distribution which is asymptotically normal $\mathcal{N}(0,n)$. Also $cov(X,Y)=cn$.

My question: What additional conditions are known to imply that the joint distribution of $(X,Y)$ is asymptotically normal?

I solved my particular application in random digraphs, but I'm wondering if I'm missing some general criteria that are not hard to apply.

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