this section, we suppose that $\{\xi(t): \ t\in T\}$, where $T$ is an arbitrary index set, is normalized to standard Fréchet margins and admits a representation $$\begin{aligned}
  \label{def_xi2}
  \xi(t) = \max_{i\in{\mathbb N}} U_i V_i(t), \quad t\in T,\end{aligned}$$ where $\sum_{i\in{\mathbb N}}\delta_{U_i}$ is a PPP on $(0,\infty)$ with intensity $u^{-2}du$, which we call *Fréchet point process* in the following. The $\{V_i\}_{i\in{\mathbb N}}$ are independent copies of a non-negative stochastic process $\{V(t): \ t\in T\}$ with $\sE V(t) =
1$, $t\in T$. Note that is slightly less restrictive than the representation in that we do not require that $V(t_0)=1$ a.s. for some $t_0\in T$. For any fixed $t_0\in T$, we have $$\begin{aligned}
  \label{decomp_V}
  \xi(t) \eqdist \max_{i\in{\mathbb N}} U_i 
  \left({\mathbf 1}_{P_i=0}V^{(1)}_i(t) + {\mathbf 1}_{P_i=1}V^{(2)}_i(t)\right), 
  \quad t\in T,\end{aligned}$$ where $\{P_i\}_{i\in{\mathbb N}}$ are i.i.d. Bernoulli variables with parameter $p=\sP(V(t_0) = 0)$ and the $V^{(1)}_i$ and $V^{(2)}_i$ are independent copies of the process $\{V(t): \ t\in T\}$, conditioned on the events $\{V(t_0) > 0\}$ and $\{V(t_0)= 0\}$, respectively.

Note that for $k\in{\mathbb N}$, $t_0, \ldots, t_k\in T$, the vector $\Xi =
(\xi(t_0),\dots,\xi(t_k))$ follows a $(k+1)$-variate extreme-value distribution and its distribution function $G$ can therefore be written as $$\begin{aligned}
  \label{def_mu}
  G(\mathbf{x}) = \exp( -\mu( [{\mathbf 0},\mathbf{x}]^C) ), 
  \quad 