standard Fréchet margins with distribution function $\exp(-1/x)$ for $x \geq 0$. In this paper, we restrict to two specific choices for $f$ and $({\mathcal S}, \mathfrak S, \nu)$ and consider processes that admit one of the resulting representations. First, let $\{W(t) : t\in T\}$ be a non-negative stochastic process with $\sE W(t) = 1$, $t\in T$, and $W(t_0) = 1$ a.s. for some point $t_0 \in T$. The latter condition means that $W(t)$ simply describes the multiplicative increment of $W$ w.r.t. the location $t_0$. For $({\mathcal S}, \mathfrak S, \nu)$ being the canonical probability space for the sample paths of $W$ and with $f(w,t)=w(t)$, $w\in{\mathcal S}$, $t\in T$, we refer to $$\begin{aligned}
  \label{def_xi}
  \xi(t) = \max_{i\in{\mathbb N}} U_i W_i(t), \quad t\in T,\end{aligned}$$ as the *incremental representation* of $\xi$, where $\{W_i\}_{i\in{\mathbb N}}$ are independent copies of $W$. Since $T$ is an arbitrary index set, the above definition covers multivariate extreme value distributions, i.e. $T=\{t_1,\dots,t_k\}$, as well as max-stable random fields, i.e. $T = {\mathbb R}^d$.\
For the second specification, let $\{F(t): \ t \in {\mathbb R}^d\}$ be a stochastic process with sample paths in the space $C({\mathbb R}^d)$ of non-negative continuous functions, such that $$\begin{aligned}
  \label{assumption_integral}
  \textstyle \sE \int_{{\mathbb R}^d}  F(t) 