On Max-Semistable Laws and Extremes for Dynamical Systems

Suppose ( f, X, µ) is a measure preserving dynamical system and φ: X → R a measurable observable. Let X_i = φ ◦ fⁱ⁻¹ denote the time series of observations on the system, and consider the maxima process M_n:= max{X₁, …, X_n }. Under linear scaling of M_n, 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.