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This paper provides the basis for new methods of inference for max-stableprocesses \xi\ on general spaces that admit a certain incrementalrepresentation, which, in important cases, has a much simpler structure thanthe max-stable process itself. A corresponding peaks-over-threshold approachwill incorporate all single events that are extreme in some sense and willtherefore rely on a substantially larger amount of data in comparison toestimation procedures based on block maxima. Conditioning a process \eta\ inthe max-domain of attraction of \xi\ on being extremal, several convergenceresults for the increments of \eta\ are proved. In a similar way, the shapefunctions of mixed moving maxima (M3) processes can be extracted from suitablyconditioned single events \eta. Connecting the two approaches, transformationformulae for processes that admit both an incremental and an M3 representationare identified.

Representations of max-stable processes based on single extreme events