Renormalization flow in extreme value statistics

TL;DR

This paper presents a PDE-based renormalization group approach to extreme value statistics, revealing fixed points, eigenfunctions, and invariant manifolds, and highlighting connections to stable distributions and the central limit theorem.

Contribution

It introduces a PDE framework for the renormalization flow in extreme value statistics, providing exact solutions and insights into distribution behaviors near fixed points.

Findings

Fixed points correspond to Fisher-Tippett distributions.

Exact solutions describe unstable manifolds near Gumbel.

Connections established between Weibull distributions and Le9vy stable distributions.

Abstract

The renormalization group transformation for extreme value statistics of independent, identically distributed variables, recently introduced to describe finite size effects, is presented here in terms of a partial differential equation (PDE). This yields a flow in function space and gives rise to the known family of Fisher-Tippett limit distributions as fixed points, together with the universal eigenfunctions around them. The PDE turns out to handle correctly distributions even having discontinuities. Remarkably, the PDE admits exact solutions in terms of eigenfunctions even farther from the fixed points. In particular, such are unstable manifolds emanating from and returning to the Gumbel fixed point, when the running eigenvalue and the perturbation strength parameter obey a pair of coupled ordinary differential equations. Exact renormalization trajectories corresponding to linear combinations of eigenfunctions can also be given, and it is shown that such are all solutions of the PDE. Explicit formulas for some invariant manifolds in the Fr\'echet and Weibull cases are also presented. Finally, the similarity between renormalization flows for extreme value statistics and the central limit problem is stressed, whence follows the equivalence of the formulas for Weibull distributions and the moment generating function of symmetric L\'evy stable distributions.