Proposition \[calculateW\], the incremental process $W$ in the representation is given by $$\begin{aligned}
    \sP(W(t_1) \leq s_1, \dots W(t_k) \leq s_k) 
    = \left(1 + \sum_{i=1}^k s_i^{-q}\right)^{1/q - 1}.
  \end{aligned}$$

Continuous sample paths
-----------------------

In this subsection, we provide an analog result to Theorem \[theo\_cond\_increments\_general\], in which convergence in the sense of finite-dimensional distributions is replaced by weak convergence on function spaces. In the following, for a Borel set $U\subset{\mathbb R}^d$, we denote by $C(U)$ and $C^+(U)$ the space of non-negative and strictly positive continuous functions on $U$, respectively, equipped with the topology of uniform convergence on compact sets.

\[theo\_cond\_increments\_cont\] Let $K$ be a compact subset of ${\mathbb R}^d$ and $\{\eta(t): \ t\in K\}$ be a process with positive and continuous sample paths in the MDA of a max-stable process $\{\xi(t): \ t\in K\}$ as in in the sense of weak convergence on $C(K)$. In particular, suppose that $$\frac 1n \max_{i=1}^n \eta_i(\cdot) \cvgdist \xi(\cdot), \quad
  n\to\infty.$$ Let $W$ be the incremental process from and $Z$ a Pareto random variable, independent of $W$. Then, for any sequence $a(n)$ of real numbers with $a(n) \to \infty$, we have the weak convergence on $(0,\infty)\times C(K)$ $$\begin{aligned}
    \left(\frac{\eta(t_0)}{a(n)}, 
    \frac{\eta(\cdot)}{\eta(t_0)} \ \Big|\ \eta(t_0) > a(n) \right)
    \cvgdist 