Electronic Journal of Probability

Cramér’s estimate for stable processes with power drift

Christophe Profeta and Thomas Simon

Full-text: Open access

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


Export citation

References

  • [1] G. E. Andrews, R. Askey and R. Roy. Special functions. Cambridge University Press, Cambridge, 1999.
  • [2] F. Aurzada and T. Kramm. The first passage time problem over a moving boundary for asymptotically stable Lévy processes. J. Theoret. Probab. 29 (3), 737-760, 2016.
  • [3] F. Aurzada and T. Simon. Persistence probabilities and exponents. In: Lévy Matters V. Functionals of Lévy processes. Lect. Notes Math. 2149, 183-221, 2015.
  • [4] J. Bertoin. Lévy processes. Cambridge University Press, Cambridge, 1996.
  • [5] J. Bertoin. Large deviations estimates in Burgers turbulence with stable noise initial data. J. Stat. Phys. 91 (3-4), 655-667, 1998.
  • [6] J. Bertoin and R. A. Doney. Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 (5), 363-365, 1994.
  • [7] K. Dȩbicki, S. Engelke and E. Hashorva. Generalized Pickands constants and stationary max-stable processes. Extremes 20 (3), 493-517, 2017.
  • [8] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi. Higher transcendental functions. Vol. I and III. McGraw-Hill, New-York, 1953.
  • [9] W. Feller. An introduction to probability theory and its applications Vol. II. Wiley, New York, 1971.
  • [10] H. Furrer. Risk processes perturbed by $\alpha $-stable Lévy motion. Scand. Actuar. J. 10, 23-35, 1998.
  • [11] P. Groeneboom and N. M. Temme. The tail of the maximum of Brownian motion minus a parabola. Elect. Comm. in Probab. 16, 458-466, 2011.
  • [12] J. Hüsler and V. Piterbarg. Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83 (2), 257-271, 1999.
  • [13] C. Klüppelberg, A. E. Kyprianou and R. A. Maller. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (4), 1766-1801, 2004.
  • [14] A. E. Kyprianou. Fluctuations of Lévy processes with applications. Introductory lectures. Springer, Heidelberg, 2014.
  • [15] W. V. Li and Q.-M. Shao. Lower tail probabilities for Gaussian processes. Ann. Probab. 32 (1), 216-242, 2004.
  • [16] C. Profeta and T. Simon. Persistence of integrated stable processes. Probab. Theory Relat. Fields 162 (3), 463-485, 2015.
  • [17] C. Profeta and T. Simon. Windings of the stable Kolmogorov process. ALEA Lat. Am. J. Probab. Math. Stat. 12 (1), 115-127, 2015.
  • [18] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall, New-York, 1994
  • [19] K. Sato. Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge, 1999.
  • [20] V. M. Zolotarev. The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Teor. Veroyat. Primenen. 9 (4), 653-662, 1964.
  • [21] V. M. Zolotarev. One-dimensional stable distributions. AMS Translations of Mathematical Monographs 65, Providence, 1986.