# CLT can be used for weighted sum of different Bernoulli variables?

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)

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