Renormalization flow in extreme value statistics

Abstract

The renormalization group transformation for extreme value statistics of independent,identically distributed variables, recently introduced to describe finite-size effects, ispresented here in terms of a partial differential equation (PDE). This yields a flow infunction space and gives rise to the known family of Fisher–Tippett limit distributions asfixed points, together with the universal eigenfunctions around them. The PDE turns outto handle correctly distributions even having discontinuities. Remarkably, the PDE admitsexact 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 pairof coupled ordinary differential equations. Exact renormalization trajectoriescorresponding to linear combinations of eigenfunctions can also be given, and it isshown that such are all solutions of the PDE. Explicit formulae for some invariantmanifolds in the Fréchet and Weibull cases are also presented. Finally, the similaritybetween renormalization flows for extreme value statistics and the central limitproblem is stressed, whence follows the equivalence of the formulae for Weibulldistributions and the moment generating function of symmetric Lévy stable distributions.