An elementary renormalization-group approach to the generalized central limit theorem and extreme value distributions

The generalized central limit theorem is a remarkable generalization of the central limit theorem, showing that the sum of a large number of independent, identically-distributed (i.i.d) random variables with infinite variance may converge under appropriate scaling to a distribution belonging to a special family known as Lévy stable distributions. Similarly, the maximum of i.i.d. variables may converge to a distribution belonging to one of three universality classes (Gumbel, Weibull and Fréchet). Here, we rederive these known results following a mathematically non-rigorous yet highly transparent renormalization-group-inspired approach that captures both of these universal results following a nearly identical procedure. (paper: classical statistical mechanics, equilibrium and non-equilibrium)