Electronic Journal of Probability

A Lévy-derived process seen from its supremum and max-stable processes

Sebastian Engelke and Jevgenijs Ivanovs

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 14, 19 pp.

Received: 11 March 2015
Accepted: 19 February 2016
First available in Project Euclid: 23 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G70: Extreme value theory; extremal processes

conditionally positive process Itô’s excursion theory mixed moving maxima representation stationary particle system Kuznetsov measure

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Engelke, Sebastian; Ivanovs, Jevgenijs. A Lévy-derived process seen from its supremum and max-stable processes. Electron. J. Probab. 21 (2016), paper no. 14, 19 pp. doi:10.1214/16-EJP1112. https://projecteuclid.org/euclid.ejp/1456246245

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