Electronic Journal of Probability
- Electron. J. Probab.
- Volume 24 (2019), paper no. 17, 21 pp.
Cramér’s estimate for stable processes with power drift
Christophe Profeta and Thomas Simon
Abstract
We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.
Article information
Source
Electron. J. Probab., Volume 24 (2019), paper no. 17, 21 pp.
Dates
Received: 26 June 2018
Accepted: 4 February 2019
First available in Project Euclid: 26 February 2019
Permanent link to this document
https://projecteuclid.org/euclid.ejp/1551150461
Digital Object Identifier
doi:10.1214/19-EJP275
Mathematical Reviews number (MathSciNet)
MR3925457
Zentralblatt MATH identifier
07055655
Subjects
Primary: 60G18: Self-similar processes 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; Lévy processes 60G52: Stable processes 60G70: Extreme value theory; extremal processes
Keywords
extremes lower tail probabilities power drift stable process
Rights
Creative Commons Attribution 4.0 International License.
Citation
Profeta, Christophe; Simon, Thomas. Cramér’s estimate for stable processes with power drift. Electron. J. Probab. 24 (2019), paper no. 17, 21 pp. doi:10.1214/19-EJP275. https://projecteuclid.org/euclid.ejp/1551150461