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I have $N$ random variables of Bernoulli distribution with different success probabilities, and I'm looking for an approximation of the distribution of their sum, when $N$ is big (but finite).

If the variables were independent, I could use the Central Limit Theorem and claim that their sum has a Normal distribution.

The $N$ variables are correlated, and have the same covariance for each pair of variables.

Is there a version of the Central Limit Theorem that could be applied in this case?

I know that there is a version for "Weak Dependence", but this is not the case for me as the covariance does not converge to $0$ as $N$ grows.

user107511
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    (1) What could "$N \nrightarrow \infty$" possibly mean in a context where you are asking about a limit theorem? (2) Given that these variables have different parameters (success probabilities), there are mathematical limitations on how correlated they can be. If the parameters vary a lot, the common covariance is going to be so small as to make no difference. What matters are the details of how the parameters vary as $N$ grows. – whuber Jul 16 '22 at 20:20
  • (1) I'm looking for an approximation for the distribution of the sum, my point was that I don't have an infinite time series as in weak dependence case, I edited that because it's probably unclear (2) What are the mathematical limitations for Bernoulli variables? (2*) I deal with multiple cases - from the case that all variables has the same success probability till the case that the success probability ranges from nearly 0 to nearly 1, the distribution of success probability does not depend on $N$, but on my starting conditions of the problem. – user107511 Jul 16 '22 at 20:31
  • The equations at https://stats.stackexchange.com/a/285008/919 give the conditions. In particular, the number $a$ depends on the correlation $\rho$ and $a$ must be small enough to make all probabilities positive. – whuber Jul 16 '22 at 22:26

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