Max-stable processes and stationary systems of Lévy particles

Abstract

We study stationary max-stable processes $\{\eta (t):t\in \mathbb{R}\}$ 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 ${\mathrm{e}}^{-u}\mathrm{d}u$, and $Y_1,Y_2,\dots$ are i.i.d.\ copies of a process $\{Y(t):t\in \mathbb{R}\}$ 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 $M_n(t)={\max }_{i=1,\dots ,n}{\xi }_i(s_n+t)$, where ${\xi }_1,{\xi }_2,\dots$ are i.i.d.\ Lévy 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 ${\max }_{i=1,\dots ,n}{Z}_i(t/\log n)$, where $Z_1,Z_2,\dots$ 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.

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