of $M$ are given by $$\begin{aligned}
& \sP(M(t_0)\leq s_0, \ldots, M(t_k)\leq s_k) \notag\\
&= \sP\left[ \Pi_0 
  \left(\left\{(u,t,f): 
  \max_{l=0}^k u f(t_l-t)/s_l > 1\right\}\right) = 0\right] \notag\\
&= \exp\left(- \int_{C({\mathbb R}^d)} \int_{{\mathbb R}^d} 
\max_{l=0}^k (f(t_l-t)/s_l)\, \rd t \, \sP_F(\rd f) \right),\label{M3_marginal}\end{aligned}$$ $t_0, \ldots, t_k\in{\mathbb R}^d$, $s_0, \ldots, s_k\geq 0$, $k\in{\mathbb N}$.

In Section \[examples\_increment\_representation\], we were interested in recovering the incremental process $W$ from processes in the MDA of a max-stable process with incremental representation. In case of M3 processes, the object of interest is clearly the distribution of the shape function $F$. Thus, in what follows, we provide the corresponding convergence results for processes $\eta$ in the MDA of an M3 process. We distinguish between processes on ${\mathbb R}^d$ with continuous sample paths and processes on a grid (${\mathbb Z}^d$). The main idea is to consider $\eta$ in the neighborhood of its own (local) maximum, conditional on this maximum being large.

Continuous Case
---------------

Let $\{\eta(t): \, t \in {\mathbb R}^d\}$ be strictly positive and in the MDA of a mixed moving maxima process $M$ in the sense of weak convergence in $C({\mathbb R}^d)$. We assume that $\eta$ is normalized such that the norming functions in are given by $c_n(t) = 