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 ? 



Is there a central limit theorem for bounded non identically distributed random variables?
*Theorem* (Billingsley, "probability and measure", example 27.4)


Let X\_i a sequence of independent, uniformly bounded random variables with zero mean, such that $\sigma(S\_n)$ goes to infinity with n. Then $S\_n/\sigma(S\_n)$ converges in law to the normalized Laplace-Gauss distribution.


This follows from the *Lindeberg triangular array theorem*. As pointed out in the other answers, the convergence can be slow. The *Bernstein inequality* may be used to bound the tail. Under the assumption of the previous theorem, for all n, we have


$$P(S\_n>t) \leq exp(-t^2/(\sigma^2(S\_n)+Ct/3))$$


where C is a bound for the |X\_i|'s.


