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 ?
Best Answer
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.