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Unimodality of the Freely Selfdecomposable Probability Laws

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Abstract

We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato’s result in (Ann. Probab. 6:523–531, 1978).

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References

  1. Barndorff-Nielsen, O.E., Thorbjørnsen, S.: Self-decomposability and Lévy processes in free probability. Bernoulli 8(3), 323–366 (2002)

    MathSciNet  MATH  Google Scholar 

  2. Belinschi, S.T., Bercovici, H.: Partially defined semigroups relative to multiplicative free convolution. Int. Math. Res. Notices, 2005(2), 65–101 (2005)

  3. Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory. Ann. Math. 149, 1023–1060 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bercovici, H., Voiculescu, D.V.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42, 733–773 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biane, P.: On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46(3), 705–718 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Collingwood, E.F., Lohwater, A.J.: The Theory of Cluster Sets. Cambridge Tracts in Mathematics and Mathematical Physics 56. Cambridge University Press, Cambridge (1966)

    Book  MATH  Google Scholar 

  7. Fritzsche, K.: From Holomorphic Functions to Complex Manifolds. Graduate Texts in Mathematics 213. Springer, Berlin (2002)

    Book  Google Scholar 

  8. Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley Publishing Company, Inc, Reading, MA (1968)

    MATH  Google Scholar 

  9. Haagerup, U., Thorbjørnsen, S.: On the free Gamma distributions. Indiana Univ. Math. J. 63(4), 1159–1194 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, H.-W.: Supports of measures in a free additive convolution semigroup. Int. Math. Res. Notices. Published online, May 8, 2014. doi:10.1093/imrn/rnu064

  11. Huang, H.-W.: Supports, Regularity and \(\boxplus \)-Infinite Divisibility for Measures of the form \((\mu ^{\boxplus {p}})^{\uplus {q}}\). arXiv:1209.5787v1

  12. Lindelöf, E.: Sur un principe général de l’analyse et ses applications à la théorie de la représentation conforme. Acta Soc. Sci. Fenn. 46(4), 1–35 (1915)

    MATH  Google Scholar 

  13. Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106, 409–438 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, 68 (1999)

  15. Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, London (1978)

    MATH  Google Scholar 

  16. Sakuma, N.: On free selfdecomposable distributions. In: Shimura, T. (ed.) Problems on Infinitely Divisible Processes, vol. 275, pp. 30–33. The Institute of Statistical Mathematics (2011)

  17. Voiculescu, D.V.: Addition of certain non-commuting random variables. J. Funct. Anal. 66, 323–346 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yamazato, M.: Unimodality of infinitely divisible distribution functions of class L. Ann. Probab. 6, 523–531 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This paper was initiated during the “Workshop on Analytic, Stochastic, and Operator Algebraic Aspects of Noncommutative Distributions and Free Probability” at the Fields Institute in July 2013. The authors would like to express their sincere gratitude for the generous support and the stimulating environment provided by the Fields Institute. The authors would also like to thank an anonymous referee for comments, which have improved the paper, and in particular for pointing out connections between our paper and Biane’s paper [5]. T. H. was supported by Marie Curie Actions—International Incoming Fellowships Project 328112 ICNCP. S.T. was partially supported by The Thiele Centre for Applied Mathematics in Natural Science at The University of Aarhus.

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  1. Takahiro Hasebe

    Present address: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo, 060-0810, Japan

Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Franche-Comté, 16 route de Gray, 2503, Besançon Cedex, France

    Takahiro Hasebe

  2. Department of Mathematics, University of Aarhus, Ny Munkegade 118, 8000, Århus C, Denmark

    Steen Thorbjørnsen

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  1. Takahiro Hasebe
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  2. Steen Thorbjørnsen
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Correspondence to Takahiro Hasebe.

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Hasebe, T., Thorbjørnsen, S. Unimodality of the Freely Selfdecomposable Probability Laws. J Theor Probab 29, 922–940 (2016). https://doi.org/10.1007/s10959-015-0595-y

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  • DOI: https://doi.org/10.1007/s10959-015-0595-y

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