---
abstract: |
    This paper provides the basis for new methods of inference for max-stable processes $\xi$ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima.\
    Conditioning a process $\eta$ in the max-domain of attraction of $\xi$ on being *extremal*, several convergence results for the increments of $\eta$ are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events $\eta$. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.
bibliography:
- 'HREstimation.bib'
title: |
    Representations of max-stable processes\
    based on single extreme events
---

Introduction
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The joint extremal behavior at multiple locations of some random process $\{\eta(t): t\in T\}$, $T$ an arbitrary index set, can be captured via its limiting *max-stable process*, assuming the latter exists and is non-trivial everywhere. Then, for independent copies $\eta_i$ of $\eta$, $i\in{\mathbb 