Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution

TL;DR

This paper analytically and numerically investigates the convergence of the block-maxima method to the Generalized Extreme Value distribution in discrete dynamical systems, highlighting conditions for robust parameter estimation.

Contribution

It provides analytical expressions for EV distribution parameters in systems with absolutely continuous invariant measures and compares them with numerical convergence results.

Findings

Robust parameter estimation in regular maps with mixing properties.

Failure of classical EV fitting in non-mixing systems.

Alternative observable functions can explain empirical distributions.

Abstract

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. [2006] have found analytical results.