most recent 30 from http://mathoverflow.net 2013-05-19T14:30:28Z http://mathoverflow.net/feeds/user/7082 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29508/is-there-a-central-limit-theorem-for-bounded-non-identically-distributed-random-v Caroline Fontaine 2010-06-25T13:02:35Z 2010-06-30T14:31:07Z

<p>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 <strong>central limit theorem</strong> 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.</p> <p>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 ? </p>