Unimodality of Hitting Times for Stable Processes

  • Julien Letemplier
  • Thomas Simon
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


We show that the hitting times for points of real α-stable Lévy processes (1 < α ≤ 2) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al. (Sémin Probab XLII:187–227, 2009), whereas in the completely asymmetric case we apply an identity of the second author (Simon, Stochastics 83(2):203–214, 2011). The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al. (Electr. J. Probab. 19:30, 1–26, 2014), for which we also provide a short independent proof.


Hitting time Kanter random variable Self-decomposability Size-bias Stable Lévy process Unimodality 



Ce travail a bénéficié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.


  1. 1.
    L. Chaumont, M. Yor, Exercises in Probability (Cambridge University Press, Cambridge, 2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    I. Cuculescu, R. Theodorescu, Multiplicative strong unimodality. Aust. New Zeal. J. Stat. 40(2), 205–214 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    W. Jedidi, T. Simon, Further examples of GGC and HCM densities. Bernoulli 19, 1818–1838 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Kanter, Stable densities under change of scale and total variation inequalities. Ann. Probab. 3, 697–707 (1975)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Kuznetsov. On the density of the supremum of a stable process. Stoch. Proc. Appl. 123(3), 986–1003 (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    A. Kuznetsov, A.E. Kyprianou, J.C. Millan, A.R. Watson, The hitting time of zero for a stable process. Electr. J. Probab. 19, Paper 30, 1–26 (2014)Google Scholar
  7. 7.
    D. Monrad. Lévy processes: Absolute continuity of hitting times for points. Z. Wahrsch. verw. Gebiete 37, 43–49 (1976)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    D. Pestana, D.N. Shanbhag, M. Sreehari. Some further results in infinite divisibility. Math. Proc. Camb. Phil. Soc. 82, 289–295 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    U. Rösler, Unimodality of passage times for one-dimensional strong Markov processes. Ann. Probab. 8(4), 853–859 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar
  11. 11.
    T. Simon, Hitting densities for spectrally positive stable processes. Stochastics 83(2), 203–214 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    T. Simon, A multiplicative short proof for the unimodality of stable densities. Elec. Comm. Probab. 16, 623–629 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    T. Simon, On the unimodality of power transformations of positive stable densities. Math. Nachr. 285(4), 497–506 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    C. Stone, The set of zeros of a semi-stable process. Ill. J. Math. 7, 631–637 (1963)zbMATHGoogle Scholar
  15. 15.
    M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes. Nagoya Math. J. 119, 143–172 (1990)MathSciNetzbMATHGoogle Scholar
  16. 16.
    K. Yano, Y. Yano, M. Yor, On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes. Sémin. Probab. XLII, 187–227 (2009)Google Scholar
  17. 17.
    V.M. Zolotarev. One-Dimensional Stable Distributions (Nauka, Moskva, 1983)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire Paul PainlevéUniversité Lille 1Villeneuve d’Ascq CedexFrance
  2. 2.Laboratoire de physique théorique et modèles statistiquesUniversité Paris SudOrsay CedexFrance

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