can be normalized such that the norming functions in become $c_n(t) = 1/n$ and $b_n(t) = 0$, $n\in{\mathbb N}$, $t\in T$ [@res2008 Prop. 5.10].

For $\mathbf{X} = (\eta(t_0),\dots,\eta(t_k))$, which is in the MDA of the random vector $\Xi=(\xi(t_0),\dots,\xi(t_k))$, it follows from [@res2008 Prop. 5.17] that $$\begin{aligned}
  \label{conv_resnick}
  \lim_{m\to\infty} m \sP( \mathbf{X}/m \in B ) = \mu(B),\end{aligned}$$ for all elements $B$ of the Borel $\sigma$-algebra $\mathcal B(E)$ of $E$ bounded away from $\{{\mathbf 0}\}$ with $\mu(\partial B)=0$, where $\mu$ is defined by . For $s_0> 0$ and ${\mathbf s}=(s_1, \ldots,
s_k)\in [0, \infty)^{k}$, we consider the sets $A_{s_0}=(s_0,\infty)\times [0, \infty)^k$, $A=A_1$ and $B_{\mathbf{s}} = \{\mathbf{x} \in [0, \infty)^{k+1} :
      (x^{(1)},\dots,x^{(k)}) \leq x^{(0)}\mathbf{s}\}$ for ${\mathbf s}$ satisfying $\sP( \Delta\tilde {\mathbf V}^{(1)}\in \partial
      [{\mathbf 0},{\mathbf s}])=0$. Then $$\begin{aligned}
  \left\{ \eta(t_0) > s_0 a(n),\, 
  \big( \eta(t_1) / \eta(t_0) ,\dots, \eta(t_k) / \eta(t_0) \big)
  \leq \mathbf{s} \right\}
  = \{ \mathbf{X} / a(n) \in B_{\mathbf{s}}\cap A_{s_0} \},\end{aligned}$$ since $B_{\mathbf{s}}$ is invariant under multiplication, i.e., $B_{\mathbf s}=cB_{\mathbf s}$ for any $c>0$. Thus, we obtain $$\begin{aligned}
  \notag
  \sP&\left( \eta(t_0) > s_0 a(n),
  \, \left( \eta(t_1) / \eta(t_0) ,\dots, 
  \eta(t_k) / \eta(t_0) \right) \leq \mathbf{s} \,\Big|\,
  \eta(t_0) > a(n) \right) \\
        \notag&= \frac{
    {a(n)} \sP( \mathbf{X} / a(n) 
    \in 