Max-stable processes and stationary systems of Lévy particles

We study stationary max-stable processes {η(t):t∈R} admitting a representation of the form η(t)=maxi∈N(Ui+Yi(t)), where ∑i=1∞δUi is a Poisson point process on R with intensity e−udu, and Y1,Y2,… are i.i.d. copies of a process {Y(t):t∈R} obtained by running a Lévy process for positive t and a dual Lévy process for negative t. We give a general construction of such Lévy–Brown–Resnick processes, where the restrictions of Y to the positive and negative half-axes are Lévy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form Mn(t)=maxi=1,…,nξi(sn+t), where ξ1,ξ2,… are i.i.d. Lévy processes and sn is a sequence such that sn∼clogn with c>0. Also, we consider maxima of the form maxi=1,…,nZi(t/logn), where Z1,Z2,… are i.i.d. Ornstein–Uhlenbeck processes driven by an α-stable noise with skewness parameter β=−1. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick (1977) to the totally skewed α-stable case.