3.  If $\xi$ admits a representation , we have $\sP(\Delta\mathbf{\tilde{V}}^{(1)}\in d \mathbf z) =
      \sP(\Delta\mathbf{{V}}\in d \mathbf z)$, which shows that is indeed a special case of Theorem \[theo\_cond\_increments\_general\].

\[rem\_thres\] In the above theorem, the sequence $a(n)$ of thresholds is only assumed to converge to $\infty$, as $n\to\infty$, ensuring that $\{\eta(t_0) > a(n)\}$ becomes a rare event. For statistical applications $a(n)$ should also be chosen such that the number of exceedances $$\begin{aligned}
    N(n) = \sum_{i=1}^n {\mathbf 1}\{ \eta_i(t_0) > a(n) \}
  \end{aligned}$$ converges to $\infty$ almost surely, where $(\eta_i)_{i\in{\mathbb N}}$ is a sequence of independent copies of $\eta$. By the Poisson limit theorem, this is equivalent to the additional assumption that $\lim_{n\to\infty} a(n)/n = 0$, since in that case $n\sP(\eta(t_0) >
  a(n)) = n / a(n) \to \infty$, as $n\to\infty$.

[@eng2012a] consider Hüsler-Reiss distributions [@hue1989; @kab2011] and obtain their limiting results by conditioning on certain extremal events $A\subset E$. They show that various choices of $A$ are sensible in the Hüsler-Reiss case, leading to different limiting distributions of the increments of $\eta$. In case $\xi$ is a Brown-Resnick process and $A =
(1,\infty)\times [0, \infty)^{k}$ the assertions of Theorem \[theo\_cond\_increments\_general\] and [@eng2012a Thm. 3.3] coincide.

A commonly used class of 