N}$, the functions $b_n: T \to {\mathbb R}$, $c_n : T\to (0,\infty)$ can be chosen such that the convergence $$\begin{aligned}
\label{MDA}
  \xi(t) = \lim_{n\to\infty} c_n(t) \Big(\max_{i=1}^n \eta_i(t) - b_n(t)\Big), 
  \quad t\in T,\end{aligned}$$ holds in the sense of finite-dimensional distributions. The process $\xi$ is said to be *max-stable* and $\eta$ is in its max-domain of attraction (MDA). The theory of max-stable processes is mainly concerned with the dependence structure while the marginals are usually assumed to be known. Even for finite-dimensional max-stable distributions, the space of possible dependence structures is uncountably infinite-dimensional and parametric models are required to find a balance between flexibility and analytical tractability [@deh2006a; @res2008].

A general construction principle for max-stable processes was provided by [@deh1984; @smi1990]: Let $\sum_{i\in{\mathbb N}} \delta_{(U_i, S_i)}$ be a Poisson point process (PPP) on $(0,\infty)\times{\mathcal S}$ with intensity measure $u^{-2}\rd u\cdot \nu(\rd s)$, where $({\mathcal S},
\mathfrak S)$ is an arbitrary measurable space and $\nu$ a positive measure on ${\mathcal S}$. Further, let $f:{\mathcal S}\times T \to [0, \infty)$ be a non-negative function with $\int_{{\mathcal S}} f(s,t) \nu(\rd s) = 1$ for all $t\in T$. Then the process $$\begin{aligned}
\xi(t) = \max_{i\in{\mathbb N}} U_i f(S_i, t), \quad t\in T,\label{constr_max_stable}\end{aligned}$$ is max-stable and has 