Sum of Bernoulli random variables


Suppose S = X_1 + X_2 + ... + X_n where X_i is a 0-1 random variable
with Pr(X_i = 1) = p_i and Pr(X_i = 0) = 1-p_i.
If all the p_i are the same, it is just a binomial random variable.
However, can we say anything when the p_i are not the same?
Furthermore, what will it be if p_i are also random variables with
some distribution?

I did some simulation and it seems that the sum of the Bernoulli
random variables with different p_i looks like a normal distribution.