On Max-Semistable Laws and Extremes for Dynamical Systems

Mark P. Holland, Alef E. Sterk
2021 Entropy  
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
more » ... 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.
doi:10.3390/e23091192 pmid:34573816 fatcat:4saep2t7trdldeav7tjzyiojqy