Séminaire de Probabilités XLVI pp 345-357

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

Unimodality of Hitting Times for Stable Processes

Chapter

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.

Keywords

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

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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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