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A Lévy-derived process seen from its supremum and max-stable processes. (English) Zbl 1336.60093
Summary: We consider a process \(Z\) on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of the supremum \(\overline Z\), its time \(T\), and the process \(Z(T+\cdot )-\overline Z\). This expression is in terms of the laws of the original and the tilted Lévy processes conditioned to stay negative and positive respectively. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.

MSC:
60G51 Processes with independent increments; Lévy processes
60G70 Extreme value theory; extremal stochastic processes
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