I have a sequence of centered independent random variables $X_i$ that are all
bounded by one in absolute value. They are not identically distributed, though.
I would like to know if the **central limit theorem** is still true
for such a sequence. Putting $S_n= X_1+...+X_n$, do we have
$$
c_n = P(\ {S_n\over\sigma(S_n)} \in [a,b] ) -
{1\over \sqrt{2\pi}}\int_a^b exp(-t2/2) dt \ \rightarrow \ 0\ ?
$$
(let's assume $\sigma(S_n)$ goes to infinity with n).
I guess it is true but I can't find a reference.

Also, what can be said from the rate of convergence of $c_n$ ? Since the $X_i$ are uniformly bounded, does $c_n$ goes to zero exponentially fast ?