Finite-size scaling in extreme statistics

TL;DR

This paper investigates how finite data set sizes affect the convergence and shape of extreme value distributions, introducing a renormalization method for iid variables and analyzing correlated systems like percolation and 1/f noise.

Contribution

It introduces a renormalization approach to characterize finite-size effects on extreme value distributions for both iid and correlated systems.

Findings

Finite-size corrections depend on the convergence exponent in iid cases.

The shape correction for correlated systems like percolation matches simulations.

For strong correlations, the correction is expressed via the limit distribution.

Abstract

We study the convergence and shape correction to the limit distributions of extreme values due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. We find that, for the correlated systems of subcritical percolation and 1/f^alpha stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f^alpha noise, the shape correction is obtained in terms of the limit distribution itself.