Extreme value distributions and Renormalization Group

TL;DR

This paper extends extreme value theory by exploring general rescalings using Renormalization Group methods, revealing new limit distributions beyond classical types and analyzing their domains of attraction.

Contribution

It introduces a novel approach employing Renormalization Group transformations to identify new limit distributions under general rescalings in extreme value theory.

Findings

Discovery of new limit distributions beyond classical types

Analysis of domains of attraction for these new distributions

Method for computing finite-size corrections

Abstract

In the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent, identically distributed random variables are studied. So far, only affine rescalings have been considered. We show, however, that more general rescalings are natural and lead to new limit distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The problem is approached using the language of Renormalization Group transformations in the space of probability densities. The limit distributions are fixed points of the transformation and the study of the differential around them allows a local analysis of the domains of attraction and the computation of finite-size corrections.