Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Chaumont, M. Yor, Exercises in Probability (Cambridge University Press, Cambridge, 2003)
I. Cuculescu, R. Theodorescu, Multiplicative strong unimodality. Aust. New Zeal. J. Stat. 40(2), 205–214 (1998)
W. Jedidi, T. Simon, Further examples of GGC and HCM densities. Bernoulli 19, 1818–1838 (2013)
M. Kanter, Stable densities under change of scale and total variation inequalities. Ann. Probab. 3, 697–707 (1975)
A. Kuznetsov. On the density of the supremum of a stable process. Stoch. Proc. Appl. 123(3), 986–1003 (2013)
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)
D. Monrad. Lévy processes: Absolute continuity of hitting times for points. Z. Wahrsch. verw. Gebiete 37, 43–49 (1976)
D. Pestana, D.N. Shanbhag, M. Sreehari. Some further results in infinite divisibility. Math. Proc. Camb. Phil. Soc. 82, 289–295 (1977)
U. Rösler, Unimodality of passage times for one-dimensional strong Markov processes. Ann. Probab. 8(4), 853–859 (1980)
K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999)
T. Simon, Hitting densities for spectrally positive stable processes. Stochastics 83(2), 203–214 (2011)
T. Simon, A multiplicative short proof for the unimodality of stable densities. Elec. Comm. Probab. 16, 623–629 (2011)
T. Simon, On the unimodality of power transformations of positive stable densities. Math. Nachr. 285(4), 497–506 (2012)
C. Stone, The set of zeros of a semi-stable process. Ill. J. Math. 7, 631–637 (1963)
M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes. Nagoya Math. J. 119, 143–172 (1990)
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)
V.M. Zolotarev. One-Dimensional Stable Distributions (Nauka, Moskva, 1983)
Acknowledgements
Ce travail a bénéficié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Letemplier, J., Simon, T. (2014). Unimodality of Hitting Times for Stable Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-11970-0_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11969-4
Online ISBN: 978-3-319-11970-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)