(Z,  W(\cdot)),
  \end{aligned}$$ as $n$ tends to $\infty$.

\[weak\_conv\_Rd\] Analogously to [@whi1970 Thm. 5], weak convergence of a sequence of probability measures $P_n$, $n\in{\mathbb N}$, to some probability measure $P$ on $C({\mathbb R}^d)$ is equivalent to weak convergence of $P_n r_j^{-1}$ to $P r_j^{-1}$ on $C([-j, j]^d)$ for all $j\geq 1$, where $r_j :
C({\mathbb R}^d) \to C([-j,j]^d)$ denotes the restriction of a function to the cube $[-j, j]^d$. Hence the assertion of Theorem \[theo\_cond\_increments\_cont\] remains valid if the compact set $K$ is replaced by ${\mathbb R}^d$.

As the process $\xi$ is max-stable and $\eta\in\text{MDA}(\xi)$, similarly to the case of multivariate max-stable distributions (cf. Theorem \[theo\_cond\_increments\_general\]), we have that $$\begin{aligned}
 \label{conv_dehaan}
  \lim_{u \to \infty} u\sP(\eta / u \in B) = \mu(B)
 \end{aligned}$$ for any Borel set $B \subset C(K)$ bounded away from $0^K$, i.e., $\inf\{\sup_{s\in K} f(s) : \ f\in B\} > 0$, and with $\mu(\partial
 B) = 0$ [@deh2006a Cor. 9.3.2], where $\mu$ is the *exponent measure* of $\xi$, defined by $$\begin{aligned}
  & \sP(\xi(s) \leq x_j, \ s \in K_j, \ j=1,\ldots,m) \nonumber \\
  &={} \exp\left[-\mu\left(\left\{ f \in C(K): \ \textstyle\sup_{s \in K_j} f(s) > x_j \textrm{ for some } j \in \{1,\ldots,m\} 
  \right\}\right)\right]
 \end{aligned}$$ for $x_j \geq 0$, 