On the ergodicity and mixing of max-stable processes

Abstract

Max-stable processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes. We do so in terms of their spectral representations by using extremal integrals. The large classes of moving maxima and mixed moving maxima processes are shown to be mixing. Other examples of ergodic doubly stochastic processes and non-ergodic processes are also given. The ergodicity conditions involve a certain measure of dependence. We relate this measure of dependence to the one of Weintraub [K.S.Weintraub, Sample and ergodic properties of some min-stable processes, Ann. Probab. 19 (2) (1991) 706–723] and show that Weintraub’s notion of ‘0-mixing’ is equivalent to mixing. Consistent estimators for the dependence function of an ergodic max-stable process are introduced and illustrated over simulated data.