\bigr),
\label{cond_incr_conv}\end{aligned}$$ $t_0,t_1,\dots,t_k\in T$, $k\in {\mathbb N}$, is established for $\eta$ being in the MDA of a Brown-Resnick process and with $W$ being the corresponding log-Gaussian random field. A similar approach exists in the theory of homogeneous discrete-time Markov chains. For instance, [@seg2007] and [@ehl2011] investigate the behavior of a Markov chain $\{M(t): t\in {\mathbb Z}\}$ conditional on the event that $M(0)$ is large. The resulting extremal process is coined the tail chain and turns out to be Markovian again. In this paper, the convergence result is generalized in different aspects. Arbitrary non-negative processes $\{W(t) : t\in T\}$ with $\sE W(t) = 1$, $t\in T$, are considered, and convergence of the conditional increments of $\eta$ in the sense of finite-dimensional distributions as well as weak convergence in continuous function spaces is shown (Theorems \[theo\_cond\_increments\_general\] and \[theo\_cond\_increments\_cont\]). Moreover, in Section \[M3representation\], similar results are established for M3 processes by considering realizations of $\eta$ around their (local) maxima. Since one and the same max-stable process $\xi$ might admit both representations and we provide formulae for switching between them in Section \[sec:switching\]. Section \[sec:application\] gives an exemplary outlook on how our results can be applied for statistical inference.

Incremental representation {#examples_increment_representation}
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Throughout 