Interestingly, the O(1/sqrt(n)) error in the CLT provided by the Berry-Eseen Theorem is the best one can do for general sequences of our RVs. However, if you ar

Interestingly, the O(1/sqrt(n)) error in the CLT provided by the Berry-Eseen Theorem is the best one can do for general sequences of our RVs. However, if you are happy to put certain 'bumps' in the normal distribution then you can improve the error to O(n^(-r/2)), where I think r is some integer depending on the smoothness of your random variables. These bumps are called an edegworth expansion.