Electronic Journal of Probability

Cramér’s estimate for stable processes with power drift

Christophe Profeta and Thomas Simon

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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


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References

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    Mathematical Reviews (MathSciNet): MR854867

The Institute of Mathematical Statistics and the Bernoulli Society

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