stationary yet non-ergodic max-stable processes on ${\mathbb R}^d$ is defined by $$\begin{aligned}
  \label{schlather_model}
  \xi(t) = \max_{i\in{\mathbb N}} U_i Y_i(t), \quad t\in{\mathbb R}^d,\end{aligned}$$ where $\sum_{i\in{\mathbb N}} \delta_{U_i}$ is a Fréchet point process, $Y_i(t)=\max(0, \tilde Y_i(t))$, $i \in {\mathbb N}$, and the $\tilde Y_i$ are i.i.d. stationary, centered Gaussian processes with $\sE(\max(0,
\tilde Y_i(t))) =1$ for all $t\in{\mathbb R}^d$ [@sch2002; @bla2011]. Note that in general, a $t_0\in{\mathbb R}^d$ s.t. $Y_i(t_0)=1$ a.s. does not exist, i.e., the process admits representation but not representation . In particular, for the extremal Gaussian process we have $p=\sP(V(t_0)=0)=1/2$ and the distribution of the increments in becomes $$\begin{aligned}
 \sP(\Delta\mathbf{\tilde{V}}^{(1)} \! \in \rd \mathbf z)
&= \frac12 \sE\Bigl[ Y(t_0) \, \Big|\,  
   (Y(t_1)/Y(t_0), \ldots, Y(t_k)/Y(t_0)) = \mathbf z, 
  \, Y(t_0)>0\Bigr]\\
  & \qquad \cdot\sP\Bigl( \bigl(Y(t_1)/Y(t_0), \ldots, 
  Y(t_k)/Y(t_0)\bigr) 
  \in \rd{\mathbf z}\, \Big|\, Y(t_0)>0 \Bigr).\end{aligned}$$

While the Hüsler-Reiss distribution is already given by the incremental representation , cf. [@kab2011], other distributions can be suitably rewritten, provided that the cumulative distribution function and hence the respective exponent measure $\mu$ is known.

\[calculateW\] Let $\Xi = (\xi(t_0),\dots,\xi(t_k))$ be a max-stable process on $T = \{t_0, \ldots, t_k \}$ with standard Fréchet margins and suppose that its exponent measure $\mu$ is concentrated on $(0, \infty) \times 