Electronic Journal of Probability
- Electron. J. Probab.
- Volume 21 (2016), paper no. 14, 19 pp.
A Lévy-derived process seen from its supremum and max-stable processes
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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G70: Extreme value theory; extremal processes
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