Generalised extreme value statistics and sum of correlated variables

TL;DR

This paper establishes a novel connection between generalized extreme value statistics and sums of correlated variables, extending classical limit theorems to correlated systems and providing insights into the emergence of Gumbel distributions.

Contribution

It introduces a mapping of extreme value statistics onto sum problems, extending the theory to non-identical, correlated variables, and interprets the Gumbel distribution with non-integer indices.

Findings

Classes of correlated variables with sum distributions matching extreme value laws

Extension of Gumbel, Fréchet, Weibull distributions to real k values

Relation between correlation length, system size, and extreme value index

Abstract

We show that generalised extreme value statistics -the statistics of the k-th largest value among a large set of random variables- can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Frechet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalisations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.