# 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)

## 1 Answer

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.