Approximations in free and classical Central Limit Theorems

We show asymptotic approximations of first and second order in the Central Limit Theorem of Free Probability. For the $n$-fold free convolution we establish errors bounds of order $o(n^{-1/2})$ and $o(n^{-1})$ depending on the existence of three or four moments. In the classical CLT the rate convergence in the entropy distance to the normal distribution and the behavior of this distance under convolution is investigated.
Joint works with G. Chistyakov and S. Bobkov.