Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Unimodality for free Lévy processes

Takahiro Hasebe and Noriyoshi Sakuma

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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897751

Digital Object Identifier
doi:10.1214/16-AIHP742

Mathematical Reviews number (MathSciNet)
MR3634280

Zentralblatt MATH identifier
1379.46051

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

Keywords
Free probability Free convolution Free Lévy process Unimodality

Citation

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. https://projecteuclid.org/euclid.aihp/1491897751


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References

  • [1] M. Anshelevich. Free martingale polynomials. J. Funct. Anal. 201 (1) (2003) 228–261.
  • [2] O. Arizmendi and T. Hasebe. On a class of explicit Cauchy–Stieltjes transforms related to monotone stable and free Poisson laws. Bernoulli 19 (5B) (2013) 2750–2767.
  • [3] O. Arizmendi and T. Hasebe. Classical and free infinite divisibility for Boolean stable laws. Proc. Amer. Math. Soc. 142 (2014) 1621–1632.
  • [4] O. Arizmendi and T. Hasebe. Classical scale mixtures of Boolean stable laws. Trans. Amer. Math. Soc. 368 (2016) 4873–4905.
  • [5] O. Arizmendi, T. Hasebe and N. Sakuma. On the law of free subordinators. ALEA Lat. Am. J. Probab. Math. Stat. 10 (1) (2013) 271–291.
  • [6] O. E. Barndorff-Nielsen and S. Thorbjørnsen. Lévy laws in free probability. Proc. Natl. Acad. Sci. USA 99 (2002) 16568–16575.
  • [7] O. E. Barndorff-Nielsen and S. Thorbjørnsen. Self-decomposability and Lévy processes in free probability. Bernoulli 8 (3) (2002) 323–366.
  • [8] S. T. Belinschi and H. Bercovici. Atoms and regularity for measures in a partially defined free convolution semigroup. Math. Z. 248 (4) (2004) 665–674.
  • [9] F. Benaych-Georges. Taylor expansions of $R$-transforms, application to supports and moments. Indiana Univ. Math. J. 55 (2) (2006) 465–481.
  • [10] H. Bercovici and V. Pata. Stable laws and domains of attraction in free probability theory. Ann. of Math. (2) 149 (1999) 1023–1060. With an appendix by P. Biane.
  • [11] H. Bercovici and D. V. Voiculescu. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (1993) 733–773.
  • [12] U. Haagerup and S. Thorbjørnsen. On the free gamma distributions. Indiana Univ. Math. J. 63 (4) (2014) 1159–1194.
  • [13] T. Hasebe. Free infinite divisibility for beta distributions and related ones. Electron. J. Probab. 19 (81) (2014) 1–33.
  • [14] T. Hasebe and N. Sakuma. Unimodality of Boolean and monotone stable distributions. Demonstratio Math. 48 (3) (2015) 424–439.
  • [15] T. Hasebe and S. Thorbjørnsen. Unimodality of the freely selfdecomposable probability laws. J. Theoret. Probab. To appear, 2016. DOI: 10.1007/s10959-015-0595-y.
  • [16] H.-W. Huang. Supports of measures in a free additive convolution semigroup. Int. Math. Res. Not. IMRN 2015 (12) (2015) 4269–4292.
  • [17] H.-W. Huang. Supports, regularity and $\boxplus$-infinite divisibility for measures of the form $(\mu^{\boxplus p})^{\uplus q}$. Available at arXiv:1209.5787v1.
  • [18] Z. J. Jurek. Relations between the $s$-selfdecomposable and selfdecomposable measures. Ann. Probab. 13 (2) (1985) 592–608.
  • [19] B. V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA, 1968.
  • [20] P. Medgyessy. On a new class of unimodal infinitely divisible distribution functions and related topics. Studia Sci. Math. Hungar. 2 (1967) 441–446.
  • [21] A. Nica and R. Speicher. On the multiplication of free $N$-tuples of noncommutative random variables. Amer. J. Math. 118 (4) (1996) 799–837.
  • [22] V. Pérez-Abreu and N. Sakuma. Free infinite divisibility of free multiplicative mixtures of the Wigner distribution. J. Theoret. Probab. 25 (1) (2012) 100–121.
  • [23] N. Saitoh and H. Yoshida. The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Statist. 21 (1) (2001) 159–170.
  • [24] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Math. 68. Cambridge University Press, Cambridge, 1999.
  • [25] F. W. Steutel and K. van Harn. Infinite Divisibility of Probability Distributions on the Real Line. Dekker, New York, 2004.
  • [26] T. Watanabe. Temporal change in distributional properties of Lévy processes. In Lévy Processes, Theory and Applications 89–107. O. E. Barndorff-Nielsen, S. I. Resnick and T. Mikosch (Eds). Birkhäuser, Basel, 2001.
  • [27] S. J. Wolfe. On the unimodality of infinitely divisible distribution functions. Z. Wahrsch. Verw. Gebiete 45 (1978) 329–335.
  • [28] M. Yamazato. Unimodality of infinitely divisible distribution functions of class $L$. Ann. Probab. 6 (1978) 523–531.