I've been told that one of the implications of the central limit theorem is that as we increase the sampling of random variables, we converge faster to a normal distribution in the center and slower out in the tails.
But this isn't immediately obvious to me. A Google search on this hardly yields any result, but I did find work on the Chernoff bound and Berry-Essen theorem that give estimates on the speed of convergence. Can someone give a clearer explanation or point me to more references on this?
Thanks!