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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Barndorff-Nielsen, O.E., Thorbjørnsen, S.: Self-decomposability and Lévy processes in free probability. Bernoulli 8(3), 323–366 (2002)
Belinschi, S.T., Bercovici, H.: Partially defined semigroups relative to multiplicative free convolution. Int. Math. Res. Notices, 2005(2), 65–101 (2005)
Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory. Ann. Math. 149, 1023–1060 (1999)
Bercovici, H., Voiculescu, D.V.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42, 733–773 (1993)
Biane, P.: On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46(3), 705–718 (1997)
Collingwood, E.F., Lohwater, A.J.: The Theory of Cluster Sets. Cambridge Tracts in Mathematics and Mathematical Physics 56. Cambridge University Press, Cambridge (1966)
Fritzsche, K.: From Holomorphic Functions to Complex Manifolds. Graduate Texts in Mathematics 213. Springer, Berlin (2002)
Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley Publishing Company, Inc, Reading, MA (1968)
Haagerup, U., Thorbjørnsen, S.: On the free Gamma distributions. Indiana Univ. Math. J. 63(4), 1159–1194 (2014)
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
Huang, H.-W.: Supports, Regularity and \(\boxplus \)-Infinite Divisibility for Measures of the form \((\mu ^{\boxplus {p}})^{\uplus {q}}\). ar**v:1209.5787v1
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)
Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106, 409–438 (1992)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, 68 (1999)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, London (1978)
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)
Voiculescu, D.V.: Addition of certain non-commuting random variables. J. Funct. Anal. 66, 323–346 (1986)
Yamazato, M.: Unimodality of infinitely divisible distribution functions of class L. Ann. Probab. 6, 523–531 (1978)
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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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