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Suppose $$ z_i \sim Bernoulli (p_i) $$

Can we use CLT for the following weighted sum?

$$ S = \sum_i w_i z_i $$

i.e. can $S$ be approximated with a normal distribution? If yes, with which theorem? (I suppose the classical CLT holds only for average of iid variables)

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Either the Lyapunov CLT or the Lindeberg CLT will be what you seek.

In each case let $X_i\,=\,w_i\,z_i$ and apply the theorems as given there to the $X_i$.

In a great many cases of the kind you suggest (and likely all that you care about), checking Lyapunov's condition should be sufficient.

However, unless you have some weird edge case, I think Lindeberg's should work in all cases of the kind you need.

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