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Classical scale mixtures of Boolean stable laws


Authors: Octavio Arizmendi and Takahiro Hasebe
Journal: Trans. Amer. Math. Soc. 368 (2016), 4873-4905
MSC (2010): Primary 46L54, 60E07
DOI: https://doi.org/10.1090/tran/6792
Published electronically: October 8, 2015
MathSciNet review: 3456164
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Abstract: We study Boolean stable laws, $ \mathbf {b}_{\alpha ,\rho }$, with stability index $ \alpha $ and asymmetry parameter $ \rho $. We show that the classical scale mixture of $ \mathbf {b}_{\alpha ,\rho }$ coincides with a free mixture and also a monotone mixture of $ \mathbf {b}_{\alpha ,\rho }$. For this purpose we define the multiplicative monotone convolution of probability measures, one supported on the positive real line and the other arbitrary.

We prove that any scale mixture of $ \mathbf {b}_{\alpha ,\rho }$ is both classically and freely infinitely divisible for $ \alpha \leq 1/2$ and also for some $ \alpha >1/2$. Furthermore, we show the multiplicative infinite divisibility of $ \mathbf {b}_{\alpha ,1}$ with respect to classical, free and monotone convolutions.

Scale mixtures of Boolean stable laws include some generalized beta distributions of the second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and Möller.

We use a representation of $ \mathbf {b}_{\alpha ,1}$ as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and Młotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that $ \mathbf {b}_{\alpha ,1}$ has free divisibility indicator 0 for $ 1/2<\alpha $.


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  • [1] Octavio Arizmendi, Statistics of blocks in $ k$-divisible non-crossing partitions, Electron. J. Combin. 19 (2012), no. 2, Paper 47, 22. MR 2946105
  • [2] Octavio Arizmendi and Takahiro Hasebe, Classical and free infinite divisibility for Boolean stable laws, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1621-1632. MR 3168468, https://doi.org/10.1090/S0002-9939-2014-12111-3
  • [3] Octavio Arizmendi and Takahiro Hasebe, On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws, Bernoulli 19 (2013), no. 5B, 2750-2767. MR 3160570, https://doi.org/10.3150/12-BEJ473
  • [4] Octavio Arizmendi and Takahiro Hasebe, Semigroups related to additive and multiplicative, free and Boolean convolutions, Studia Math. 215 (2013), no. 2, 157-185. MR 3071490, https://doi.org/10.4064/sm215-2-5
  • [5] Octavio Arizmendi, Takahiro Hasebe, and Noriyoshi Sakuma, On the law of free subordinators, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 271-291. MR 3083927
  • [6] Octavio Arizmendi E. and Victor Pérez-Abreu, The $ S$-transform of symmetric probability measures with unbounded supports, Proc. Amer. Math. Soc. 137 (2009), no. 9, 3057-3066. MR 2506464 (2010g:46107), https://doi.org/10.1090/S0002-9939-09-09841-4
  • [7] Drew Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 (2011c:05001), https://doi.org/10.1090/S0065-9266-09-00565-1
  • [8] T. Banica, S. T. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), no. 1, 3-37. MR 2779129 (2011m:46121), https://doi.org/10.4153/CJM-2010-060-6
  • [9] Ole E. Barndorff-Nielsen and Steen Thorbjørnsen, Lévy laws in free probability, Proc. Natl. Acad. Sci. USA 99 (2002), no. 26, 16568-16575. MR 1947756 (2003j:46101a), https://doi.org/10.1073/pnas.232052399
  • [10] Ole E. Barndorff-Nielsen and Steen Thorbjørnsen, Self-decomposability and Lévy processes in free probability, Bernoulli 8 (2002), no. 3, 323-366. MR 1913111 (2003c:60031)
  • [11] S. T. Belinschi and H. Bercovici, Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Not. 2 (2005), 65-101. MR 2128863 (2006f:46061), https://doi.org/10.1155/IMRN.2005.65
  • [12] Serban T. Belinschi, Marek Bożejko, Franz Lehner, and Roland Speicher, The normal distribution is $ \boxplus $-infinitely divisible, Adv. Math. 226 (2011), no. 4, 3677-3698. MR 2764902 (2011m:46122), https://doi.org/10.1016/j.aim.2010.10.025
  • [13] Serban T. Belinschi and Alexandru Nica, On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana Univ. Math. J. 57 (2008), no. 4, 1679-1713. MR 2440877 (2009f:46087), https://doi.org/10.1512/iumj.2008.57.3285
  • [14] Hari Bercovici, Multiplicative monotonic convolution, Illinois J. Math. 49 (2005), no. 3, 929-951 (electronic). MR 2210269 (2006k:46103)
  • [15] Hari Bercovici and Vittorino Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149 (1999), no. 3, 1023-1060. With an appendix by Philippe Biane. MR 1709310 (2000i:46061), https://doi.org/10.2307/121080
  • [16] Hari Bercovici and Dan Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733-773. MR 1254116 (95c:46109), https://doi.org/10.1512/iumj.1993.42.42033
  • [17] Lennart Bondesson, Generalized gamma convolutions and related classes of distributions and densities, Lecture Notes in Statistics, vol. 76, Springer-Verlag, New York, 1992. MR 1224674 (94g:60031)
  • [18] Marek Bożejko and Takahiro Hasebe, On free infinite divisibility for classical Meixner distributions, Probab. Math. Statist. 33 (2013), no. 2, 363-375. MR 3158562
  • [19] Paul H. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1980), no. 2, 171-180. MR 583216 (81i:05018), https://doi.org/10.1016/0012-365X(80)90033-3
  • [20] Uwe Franz, Monotone and Boolean convolutions for non-compactly supported probability measures, Indiana Univ. Math. J. 58 (2009), no. 3, 1151-1185. MR 2541362 (2010k:46065), https://doi.org/10.1512/iumj.2009.58.3578
  • [21] Uffe Haagerup and Sören Möller, The law of large numbers for the free multiplicative convolution, Operator algebra and dynamics, Springer Proc. Math. Stat., vol. 58, Springer, Heidelberg, 2013, pp. 157-186. MR 3142036, https://doi.org/10.1007/978-3-642-39459-1_8
  • [22] Uffe Haagerup and Hanne Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra, Math. Scand. 100 (2007), no. 2, 209-263. MR 2339369 (2008m:46139)
  • [23] Takahiro Hasebe, Free infinite divisibility for beta distributions and related ones, Electron. J. Probab. 19 (2014), no. 81, 33. MR 3256881, https://doi.org/10.1214/EJP.v19-3448
  • [24] Takahiro Hasebe, Monotone convolution and monotone infinite divisibility from complex analytic viewpoint, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 1, 111-131. MR 2646794 (2011i:60036), https://doi.org/10.1142/S0219025710003973
  • [25] Takahiro Hasebe and Alexey Kuznetsov, On free stable distributions, Electron. Commun. Probab. 19 (2014), no. 56, 12. MR 3254735, https://doi.org/10.1214/ECP.v19-3443
  • [26] Takahiro Hasebe and Noriyoshi Sakuma, Unimodality of Boolean and monotone stable distributions, Demonstr. Math. 48 (2015), no. 3, 424-439. MR 3391380, https://doi.org/10.1515/dema-2015-0030
  • [27] T. Hasebe and S. Thorbjørnsen, Unimodality of the freely selfdecomposable probability laws, J. Theoret. Probab., to appear, DOI 10.1007/s10959-015-0595-y.
  • [28] Melanie Hinz and Wojciech Młotkowski, Free powers of the free Poisson measure, Colloq. Math. 123 (2011), no. 2, 285-290. MR 2811179 (2012k:46078), https://doi.org/10.4064/cm123-2-11
  • [29] H.-W. Huang, Supports, regularity and $ \boxplus $-infinite divisibility for measures of the form $ (\mu ^{\boxplus p})^{\uplus q}$, arXiv:1209.5787v1
  • [30] Wissem Jedidi and Thomas Simon, Further examples of GGC and HCM densities, Bernoulli 19 (2013), no. 5A, 1818-1838. MR 3129035, https://doi.org/10.3150/12-BEJ431
  • [31] Zbigniew J. Jurek, Relations between the $ s$-self-decomposable and self-decomposable measures, Ann. Probab. 13 (1985), no. 2, 592-608. MR 781426 (87a:60016)
  • [32] A. Ya. Khintchine, On unimodal distributions, Izv.Nauk Mat. i Mekh. Inst. Tomsk 2 (1938), 1-7 (in Russian).
  • [33] Hans Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (1992), no. 2, 409-438. MR 1165862 (94g:46069), https://doi.org/10.1016/0022-1236(92)90055-N
  • [34] Naofumi Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 3, 337-371. MR 2016316 (2005h:46093), https://doi.org/10.1142/S0219025703001365
  • [35] Alexandru Nica and Roland Speicher, A ``Fourier transform'' for multiplicative functions on non-crossing partitions, J. Algebraic Combin. 6 (1997), no. 2, 141-160. MR 1436532 (98i:46070), https://doi.org/10.1023/A:1008643104945
  • [36] Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879 (2008k:46198)
  • [37] Alexandru Nica and Roland Speicher, On the multiplication of free $ N$-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), no. 4, 799-837. MR 1400060 (98i:46069)
  • [38] Victor Pérez-Abreu and Noriyoshi Sakuma, Free infinite divisibility of free multiplicative mixtures of the Wigner distribution, J. Theoret. Probab. 25 (2012), no. 1, 100-121. MR 2886381, https://doi.org/10.1007/s10959-010-0288-5
  • [39] N. Raj Rao and Roland Speicher, Multiplication of free random variables and the $ S$-transform: the case of vanishing mean, Electron. Comm. Probab. 12 (2007), 248-258. MR 2335895 (2008f:46082), https://doi.org/10.1214/ECP.v12-1274
  • [40] K. Sato, Lévy Processes and infinitely divisible distributions, Cambridge Studies in Advanced Math. 68, 1999.
  • [41] Roland Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88. MR 1407898 (98i:46071), https://doi.org/10.1090/memo/0627
  • [42] Roland Speicher and Reza Woroudi, Boolean convolution, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 267-279. MR 1426845 (98b:46084)
  • [43] Richard P. Stanley, Parking functions and noncrossing partitions, Electron. J. Combin. 4 (1997), no. 2, Research Paper 20, approx. 14 pp. (electronic). The Wilf Festschrift (Philadelphia, PA, 1996). MR 1444167 (98m:05011)
  • [44] Fred W. Steutel and Klaas van Harn, Infinite divisibility of probability distributions on the real line, Monographs and Textbooks in Pure and Applied Mathematics, vol. 259, Marcel Dekker, Inc., New York, 2004. MR 2011862 (2005j:60004)
  • [45] Gabriel H. Tucci, Limits laws for geometric means of free random variables, Indiana Univ. Math. J. 59 (2010), no. 1, 1-13. MR 2666470 (2011m:46123), https://doi.org/10.1512/iumj.2010.59.3775
  • [46] Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323-346. MR 839105 (87j:46122), https://doi.org/10.1016/0022-1236(86)90062-5
  • [47] Dan Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), no. 2, 223-235. MR 915507 (89b:46076)
  • [48] Dan Voiculescu, Symmetries of some reduced free product $ C^\ast $-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556-588. MR 799593 (87d:46075), https://doi.org/10.1007/BFb0074909
  • [49] Jiun-Chau Wang, Strict limit types for monotone convolution, J. Funct. Anal. 262 (2012), no. 1, 35-58. MR 2852255 (2012j:46098), https://doi.org/10.1016/j.jfa.2011.09.004
  • [50] V. M. Zolotarev, One-dimensional stable distributions, Translations of Mathematical Monographs, vol. 65, American Mathematical Society, Providence, RI, 1986. Translated from the Russian by H. H. McFaden; Translation edited by Ben Silver. MR 854867 (87k:60002)

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Additional Information

Octavio Arizmendi
Affiliation: Department of Probability and Statistics, CIMAT, Guanajuato, Mexico
Email: octavius@cimat.mx

Takahiro Hasebe
Affiliation: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kitaku, Sapporo 060-0810, Japan
Email: thasebe@math.sci.hokudai.ac.jp

DOI: https://doi.org/10.1090/tran/6792
Keywords: Free convolution, Boolean stable law, infinite divisibility, mixtures, free Bessel law
Received by editor(s): May 27, 2014
Published electronically: October 8, 2015
Additional Notes: The second author was supported by Marie Curie Actions – International Incoming Fellowships (Project 328112 ICNCP) at University of Franche-Comté and also by the Global COE program “Fostering top leaders in mathematics – broadening the core and exploring new ground” at Kyoto University.
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