Open Access
Translator Disclaimer
2016 A Lévy-derived process seen from its supremum and max-stable processes
Sebastian Engelke, Jevgenijs Ivanovs
Electron. J. Probab. 21: 1-19 (2016). DOI: 10.1214/16-EJP1112


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.


Download Citation

Sebastian Engelke. Jevgenijs Ivanovs. "A Lévy-derived process seen from its supremum and max-stable processes." Electron. J. Probab. 21 1 - 19, 2016.


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

zbMATH: 1336.60093
MathSciNet: MR3485356
Digital Object Identifier: 10.1214/16-EJP1112

Primary: 60G51
Secondary: 60G70

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


Vol.21 • 2016
Back to Top