The Central Limit Theorem for Free Additive Convolution

Let {X_n}^∞_n=1be a sequence of free, identically distributed random variables with common distributionμ. Then there exist sequences {B_n}^∞_n=1and {A_n}^∞_n=1of positive and real numbers, respectively, such that sequence of random variables[formula]converges in distribution to the semicircle law if and only if the function[formula]is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik.