Extreme Values of Infinite-Measure Processes

TL;DR

This paper investigates the extreme value statistics of non-stationary, infinite-measure dynamical systems, revealing how they differ from classical universality classes and how extremes can inform about the underlying infinite invariant measure.

Contribution

It introduces a theoretical framework for understanding extremes in infinite ergodic systems, linking them to the return exponent and infinite invariant measure, with practical illustrations.

Findings

Extreme value statistics depend on the return exponent and infinite measure.

Classical extreme value universality classes do not apply to these systems.

Measurement of extremes can reveal the structure of the infinite invariant density.

Abstract

We study the statistics of the maximum and minimum of a set of $N$ random variables whose dynamical and statistical properties fall within the scope of infinite ergodic theory. These non-stationary yet recurrent systems are described, in the long-time limit, by a non-normalizable infinite invariant density. Extreme events in such systems emerge in a joint limit where the observation time $t$ is long and the number of variables $N$ is large. We show that the resulting extreme value statistics are controlled by the return exponent $\alpha$ and the infinite invariant measure, and therefore depart from the classical Fr\'echet, Gumbel, and Weibull universality classes. We illustrate the theory for weakly chaotic intermittent maps, overdamped diffusion in an asymptotically flat potential, and a stochastic model of sub-recoil laser cooling, and show how measurements of extremes can be used to infer the infinite-density structure.