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

We study stationary max-stable processes $\{\eta (t)\text{\hspace{0.17em}⁣}:t\in \mathbb{R}\}$ 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 $\{Y(t)\text{\hspace{0.17em}⁣}:t\in \mathbb{R}\}$ 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 $M_n(t)={\max }_{i=1,\dots ,n}{\xi }_i(s_n+t)$, where ${\xi }_1,{\xi }_2,\dots$ 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 ${\max }_{i=1,\dots ,n}{Z}_i(t\mathrm{/}\log n)$, 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.