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On Max-Semistable Laws and Extremes for Dynamical Systems.

Mark P Holland1, Alef E Sterk2

  • 1College of Engineering, Mathematics and Physical Sciences, Harrison Building, Streatham Campus, University of Exeter, North Park Road, Exeter EX4 4QF, UK.

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Summary
This summary is machine-generated.

This study introduces a new parametric distribution for analyzing extreme values in dynamical systems when standard assumptions are unmet. It demonstrates a max-semistable limit for scaled maxima processes in specific systems.

Keywords:
dynamical systemsextremal indexextreme value theorymax-semistable lawstail index

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Area of Science:

  • Dynamical Systems Theory
  • Extreme Value Theory
  • Probability Theory

Background:

  • Measure preserving dynamical systems generate time series data.
  • Extreme value analysis typically relies on the Generalized Extreme Value distribution.
  • This distribution requires specific regularity conditions on observables and measure density.

Purpose of the Study:

  • To explore an alternative parametric distribution for extreme value modeling.
  • To address limitations when observables or measure densities lack regular variation assumptions.
  • To investigate the asymptotic behavior of maxima processes in dynamical systems.

Main Methods:

  • Analysis of max-semistable processes as a limiting distribution.
  • Focus on piecewise uniformly expanding dynamical systems.
  • Examination of the scaled maxima process (Mn).

Main Results:

  • Identified a max-semistable distribution as a suitable model for extreme behavior.
  • Demonstrated that a max-semistable limit holds for the scaled maxima process.
  • Provided an alternative to the Generalized Extreme Value distribution under relaxed conditions.

Conclusions:

  • The max-semistable distribution offers a robust approach to extreme value analysis in dynamical systems.
  • This finding is particularly relevant for systems with non-regular observables or measure densities.
  • The study advances the understanding of extreme events in complex systems.