On Max-Semistable Laws and Extremes for Dynamical Systems

Mark P. Holland*, Alef E. Sterk

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

Suppose ( f, X, µ) is a measure preserving dynamical system and φ: X → R a measurable observable. Let Xi = φ ◦ fi−1 denote the time series of observations on the system, and consider the maxima process Mn:= max{X1, …, Xn }. Under linear scaling of Mn, its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process.

Original languageEnglish
Article number1192
Number of pages15
JournalEntropy
Volume23
Issue number9
DOIs
Publication statusPublished - Sept-2021

Keywords

  • Dynamical systems
  • Extremal index
  • Extreme value theory
  • Max-semistable laws
  • Tail index

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