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 $$ c_n = P(\ {S_n\over\sigma(S_n)} \in [a,b] ) - {1\over \sqrt{2\pi}}\int_a^b exp(-t^2/2) dt \ \rightarrow \ 0\ ? $$ (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 ?