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Unimodality of Hitting Times for Stable Processes

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2123)

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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Acknowledgements

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

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Correspondence to Julien Letemplier .

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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

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