Extreme Value Laws for Superstatistics

TL;DR

This paper explores how different superstatistics classes influence the extreme value distributions of stochastic processes, revealing that certain superstatistics lead to specific classical extreme value laws.

Contribution

It establishes the relationship between superstatistics universality classes and classical extreme value distributions, clarifying which superstatistics lead to Gumbel, Fréchet, or Weibull laws.

Findings

Weibull distribution cannot occur if the local equilibrium distribution is unbounded.

Chi-squared superstatistics typically results in Fréchet extreme value distribution.

Inverse chi-squared and lognormal superstatistics lead to Gumbel extreme value distribution.

Abstract

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fr\'echet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely $\chi^2$-superstatistics, inverse $\chi^2$-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that $\chi^2$-superstatistics generally leads an extreme value statistics described by a Fr\'echet distribution, whereas inverse $\chi^2$-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.