Example of a Non-standard Extreme Value Law

TL;DR

This paper presents an example of a dynamical system, specifically irrational circle rotations, that exhibits a non-standard extreme value distribution, contrasting with the typical Gumbel, Fréchet, or Weibull laws for well-mixing systems.

Contribution

It introduces a specific example of a non-mixing dynamical system with a non-standard limiting distribution for extreme values, expanding understanding of extreme value laws.

Findings

Irrational circle rotations can have non-standard extreme value distributions.

The limiting distribution is a step function dependent on a sequence of convergents.

Standard extreme value laws do not apply to this non-mixing system.

Abstract

It has been shown that sufficiently well mixing dynamical systems with positive entropy have extreme value laws which in the limit converge to one of the three standard distributions known for i.i.d. processes, namely Gumbel, Fr\'echet and Weibull distributions. In this short note we give an example which has a non-standard limiting distribution for its extreme values. Rotations of the circle by irrational numbers are used and it will be shown that the limiting distribution is a step function where the limit has to be taken along a suitable sequence given by the convergents.