Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Unimodality for free Lévy processes

Takahiro Hasebe and Noriyoshi Sakuma

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We will prove that: (1) A symmetric free Lévy process is unimodal if and only if its free Lévy measure is unimodal; (2) Every free Lévy process with boundedly supported Lévy measure is unimodal in sufficiently large time. (2) is completely different property from classical Lévy processes. On the other hand, we find a free Lévy process such that its marginal distribution is not unimodal for any time $s>0$ and its free Lévy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Lévy 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 Lévy process in terms of Lévy–Khintchine representation.


Nous montrons que: (1) Un processus de Lévy libre symétrique est unimodal si et seulement si sa mesure de Lévy libre est unimodale; (2) Chaque processus de Lévy libre avec mesure de Lévy à support borné est unimodal en temps suffisamment grand. (2) est une propriété tout à fait différente des processus de Lévy classiques. D’autre part, nous trouvons un processus de Lévy libre tel que la distribution marginale n’est pas unimodale pour tout temps $s>0$ et dont la mesure de Lévy libre n’est pas de support borné. Par conséquent, nous concluons que l’hypothèse sur le support de la mesure de Lévy libre dans (2) ne peut pas être supprimée. Pour la preuve, nous caractérisons (presque) l’existence d’atomes et l’existence de densités de probabilité continues pour les distributions marginales d’un processus libre Lévy en termes de sa représentation de Lévy–Khintchine.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 916-936.

Received: 6 August 2015
Revised: 5 December 2015
Accepted: 27 January 2016
First available in Project Euclid: 11 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras 60G51: Processes with independent increments; Lévy processes

Free probability Free convolution Free Lévy process Unimodality


Hasebe, Takahiro; Sakuma, Noriyoshi. Unimodality for free Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 916--936. doi:10.1214/16-AIHP742.

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