An RG approach to the generalized central limit Theorem and to extreme value distributions.
The central limit theorem is standard part of physics and maths curriculum, with important implications to all branches of science. Its generalization to scenarios where a sum of variables drawn from a heavy-tailed distribution – with infinite mean or variance – is also applicable to numerous problems, including anomalous diffusion. Another fundamental problem regards the maxima (or minima) of a large number of independent variables. Also in this case universal behavior emerges, with a family of limiting distributions emerging. I will derive the Levy stable distributions that generalize the familiar Gaussian of the central limit theorem, and the so-called extreme value distributions associated with the extremum of iid variables. Both derivations will rely on the self-similarity of the resulting distributions under appropriate transformations, in a manner reminiscent of renormalization group approaches.