\mathbf{x} \in {\mathbb R}^{k+1},\end{aligned}$$ where $\mu$ is a measure on $E = [0,\infty)^{k+1}\setminus\{{\mathbf 0}\}$, the so-called *exponent measure* of $G$ [@res2008 Prop. 5.8], and $[{\mathbf 0},\mathbf{x}]^C = E\setminus
  [{\mathbf 0},\mathbf{x}]$.

The following convergence result provides the theoretical foundation for statistical inference based on the incremental process $V$.

\[theo\_cond\_increments\_general\] Let $\{\eta(t): \ t\in T\}$ be non-negative and in the MDA of some max-stable process $\xi$ that admits a representation and suppose that $\eta$ is normalized such that holds with $c_n(t) = 1/n$ and $b_n(t) = 0$ for $n\in{\mathbb N}$ and $t\in T$. Let $a(n)\to\infty$ as $n\to\infty$. For $k\in {\mathbb N}$ and $t_0,\dots,t_k\in T$ we have the convergence in distribution on ${\mathbb R}^{k+1}$ $$\begin{aligned}
    \left(\frac{\eta(t_0)}{a(n)}, \frac{\eta(t_1)}{\eta(t_0)} ,\dots, 
    \frac{\eta(t_k)}{\eta(t_0)} 
    \ \Bigg|\ 
    \eta(t_0) > a(n)\right) \cvgdist
    \left(Z, \Delta\mathbf{\tilde{V}}^{(1)}\right),\quad n\to \infty,
  \end{aligned}$$ where the distribution of $\Delta\mathbf{\tilde{V}}^{(1)}$ is given by $$\begin{aligned}
    \sP(\Delta\mathbf{\tilde{V}}^{(1)}\in d \mathbf z) = 
    (1-p)\sP(\Delta\mathbf V^{(1)}\in d \mathbf z)
    \sE\bigl( V^{(1)}(t_0) \big| 
    \Delta\mathbf V^{(1)}=\mathbf z \bigr), \quad \mathbf{z} \geq {\mathbf 0}.
    \label{density_increment}
  \end{aligned}$$ Here, $\Delta\mathbf V^{(1)}$ denotes the vector of increments $\left(\frac{V^{(1)}(t_1)}{V^{(1)}(t_0)}, \ldots,
  \frac{V^{(1)}(t_k)}{V^{(1)}(t_0)}\right)$ with respect to $t_0$, and $Z$ is an independent Pareto variable.

Note that any process $\eta$ that satisfies the convergence in for a process $\xi$ with standard Fréchet margins 