1 / n$ and $b_n(t) = 0$, for any $n\in{\mathbb N}$ and $t\in{\mathbb R}^d$. Further suppose that the shape function $F$ of $M$ is sample-continuous and satisfies $$\begin{aligned}
  \begin{split}
    F({\mathbf}0) &= \lambda \quad a.s., \\
    F(t) & \in [0,\lambda) \ \forall t \in {\mathbb R}^d \setminus \{{\mathbf}0\} \quad a.s.
      \label{eq:Fmaxatorigin}
  \end{split}\end{aligned}$$ for some $\lambda > 0$ and $$\int_{{\mathbb R}^d} \sE\left\{  \max_{t_0 \in K} F(t_0 - t) \right\} \sd t < \infty 
  \label{eq:sup-integrability-cont}$$ for any compact set $K \subset {\mathbb R}^d$. Under these assumptions, there is an analog result to Theorem \[theo\_cond\_increments\_cont\].

\[thm\_conv\_mmm\] Let ${Q}, K \subset {\mathbb R}^d$ be compact such that $\partial {Q}$ is a Lebesgue null set and let $$\tau_{Q}: \ C({Q}) \to {\mathbb R}^d, \ f \mapsto \inf\left( \operatorname*{arg\,max}_{t \in {Q}}
 f(t) \right),$$ where [“inf”]{} is understood in the lexicographic sense. Then, under the above assumptions, for any Borel set $B \subset C(K)$ with $\sP(F / \lambda \in \partial B) = 0$, and any sequence $a(n)$ with $a(n) \to \infty$ as $n \to \infty$, we have $$\begin{aligned}
  & \lim_{\substack{\{{\mathbf}0\} \in L \nearrow {\mathbb R}^d\\ {\rm compact}}}
   \limsup_{n \to \infty} \sP\Big\{ \eta\big(\tau_{Q}(\eta|_{Q})+\cdot\big) \big/  \eta(\tau_{Q}(\eta|_{Q})) \in B \ \Big| \\[-1em]
   &  \hspace{2.5cm} \max_{t\in {Q}}\eta(t) = \max_{t \in {Q}\oplus L} 