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Unimodality for free Levy processes

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Title: Unimodality for free Levy processes
Authors: Hasebe, Takahiro Browse this author →KAKEN DB
Sakuma, Noriyoshi Browse this author
Keywords: Free probability
Free convolution
Free Levy process
Unimodality
Issue Date: May-2017
Publisher: Institute of Mathematical Statistics
Journal Title: Annales de l'I.H.P. Probabilités et statistiques
Volume: 53
Issue: 2
Start Page: 916
End Page: 936
Publisher DOI: 10.1214/16-AIHP742
Abstract: We will prove that: (1) A symmetric free Levy process is unimodal if and only if its free Levy measure is unimodal; (2) Every free Levy process with boundedly supported Levy measure is unimodal in sufficiently large time. (2) is completely different property from classical Levy processes. On the other hand, we find a free Levy process such that its marginal distribution is not unimodal for any time s > 0 and its free Levy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Levy measure in (2) cannot be dropped. For the proof we will (almost) characterize the existence of atoms and the existence of continuous probability densities of marginal distributions of a free Levy process in terms of Levy-Khintchine representation.
Type: article
URI: http://hdl.handle.net/2115/66291
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 長谷部 高広

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