$K_j \subset K$ compact. Thus, $\mu$ equals the intensity measure of the Poisson point process $\sum_{i \in {\mathbb N}}
 \delta_{U_i W_i(\cdot)}$. For $z>0$ and $D\subset C(K)$ Borel, we consider the sets $$\begin{aligned}
A_{z} &= \{f \in C(K): \ f(t_0) > z\}\\
B_D &= \{f \in C(K) : f(\cdot)/f(t_0)\in D\}\end{aligned}$$ and $A=A_1$. Note that $B_D$ is invariant w.r.t. multiplication by any positive constant. Then, as $W(t_0) = 1$ a.s., we have $\mu(A_{z}) = \int_{z}^\infty u^{-2} \sd u = z^{-1}$ and for $s_0\geq 1$ and any Borel set $D \subset C(K)$ with $\sP(W \in \partial D) = 0$, by , we get $$\begin{aligned}
  &\sP\left\{\eta(t_0) / a(n) > s_0,\ \eta(\cdot)/\eta(t_0) \in D \, \Big|\, \eta(t_0) > a(n) \right\}\\
  &= \frac{a(n) \sP\bigl\{\eta(\cdot) / a(n) \in A_{s_0} \cap B_D \cap A\bigr\}}{a(n) \sP\bigl\{\eta(\cdot) / a(n) \in A\bigr\}}\\
  &\stackrel{n \to \infty}{\longrightarrow}{} \frac{\mu(B_D \cap A_{s_0})}{\mu(A)}\\
 &={} \int_{s_0}^\infty u^{-2}\sP\bigl\{u W(\cdot) \in B_D\bigr\} \sd u\\
   &={} s_0^{-1} \sP\bigl\{W(\cdot) \in D\bigr\}, 
 \end{aligned}$$ which is the joint distribution of $Z$ and $W(\cdot)$.

\[BRproc\] For $T={\mathbb R}^d$, $d\geq 1$, let $\{Y(t): \ t\in T\}$ be a centered Gaussian process with stationary increments, continuous sample paths and $Y(t_0) = 0$ for some $t_0\in{\mathbb R}^d$. Note that by [@adl2007 Thm. 1.4.1] it is sufficient for 