Renormalization flow in extreme value statistics

Eric Bertin, Géza Györgyi

Research output: Contribution to journalArticle

10 Citations (Scopus)

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 formulae for some invariant manifolds in the Fréchet 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 formulae for Weibull distributions and the moment generating function of symmetric Lévy stable distributions.

Original languageEnglish
Article numberP08022
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2010
Issue number8
DOIs
Publication statusPublished - Jan 1 2010

Keywords

  • Extreme value statistics
  • Renormalization group

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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