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, 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 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 formulas for Weibull distributions and the moment generating function of symmetric Lévy stable distributions.

21 pages, 9 figures. Several typos and an upload error corrected. Accepted for publication in JSTAT

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• Real-Space Renormalization for disordered systems at the level of Large Deviations
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• Extreme statistics of the excitations in the random transverse Ising chain
• Renormalization flow for extreme value statistics of random variables raised to a varying power
• General limit distributions for sums of random variables with a matrix product representation
• Neutrino mass and Extreme Value Distributions in $\beta$-decay
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value