Max-stable processes and stationary systems of L\'evy particles

Abstract

We study stationary max-stable processes $\{\eta(t)\colon t\in\mathbb R\}$ admitting a representation of the form $\eta(t)=\max_{i\in\mathbb N}(U_i+ Y_i(t))$, where $\sum_{i=1}^{\infty} \delta_{U_i}$ is a Poisson point process on $\mathbb R$ with intensity ${\rm e}^{-u} {\rm d} u$, and $Y_1,Y_2,\ldots$ are i.i.d.\ copies of a process $\{Y(t)\colon t\in\mathbb R\}$ obtained by running a L\'evy process for positive $t$ and a dual L\'evy process for negative $t$. We give a general construction of such L\'evy-Brown-Resnick processes, where the restrictions of $Y$ to the positive and negative half-axes are L\'evy 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 $M_n(t)=\max_{i=1,\ldots,n} \xi_i(s_n+t)$, where $\xi_1,\xi_2,\ldots$ are i.i.d.\ L\'evy processes and $s_n$ is a sequence such that $s_n\sim c \log n$ with $c>0$. Also, we consider maxima of the form $\max_{i=1,\ldots,n} Z_i(t/\log n)$, where $Z_1,Z_2,\ldots$ are i.i.d.\ Ornstein--Uhlenbeck processes driven by an $\alpha$-stable noise with skewness parameter $\beta=-1$. 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 [Extreme values of independent stochastic processes, J.\ Appl.\ Probab., 14 (1977), pp.\ 732--739] to the totally skewed $\alpha$-stable case.