Classical scale mixtures of Boolean stable laws

Arizmendi, Octavio; Hasebe, Takahiro

doi:10.1090/tran/6792

Classical scale mixtures of Boolean stable laws
HTML articles powered by AMS MathViewer

by Octavio Arizmendi and Takahiro Hasebe PDF

Trans. Amer. Math. Soc. 368 (2016), 4873-4905 Request permission

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 $1/2<\alpha$.

References

Octavio Arizmendi, Statistics of blocks in $k$-divisible non-crossing partitions, Electron. J. Combin. 19 (2012), no. 2, Paper 47, 22. MR 2946105
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, DOI 10.1090/S0002-9939-2014-12111-3
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, DOI 10.3150/12-BEJ473
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, DOI 10.4064/sm215-2-5
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
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, DOI 10.1090/S0002-9939-09-09841-4
Drew Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274, DOI 10.1090/S0065-9266-09-00565-1
T. Banica, S. T. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), no. 1, 3–37. MR 2779129, DOI 10.4153/CJM-2010-060-6
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, DOI 10.1073/pnas.232052399
Ole E. Barndorff-Nielsen and Steen Thorbjørnsen, Self-decomposability and Lévy processes in free probability, Bernoulli 8 (2002), no. 3, 323–366. MR 1913111
S. T. Belinschi and H. Bercovici, Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Not. 2 (2005), 65–101. MR 2128863, DOI 10.1155/IMRN.2005.65
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, DOI 10.1016/j.aim.2010.10.025
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, DOI 10.1512/iumj.2008.57.3285
Hari Bercovici, Multiplicative monotonic convolution, Illinois J. Math. 49 (2005), no. 3, 929–951. MR 2210269
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, DOI 10.2307/121080
Hari Bercovici and Dan Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733–773. MR 1254116, DOI 10.1512/iumj.1993.42.42033
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, DOI 10.1007/978-1-4612-2948-3
Marek Bożejko and Takahiro Hasebe, On free infinite divisibility for classical Meixner distributions, Probab. Math. Statist. 33 (2013), no. 2, 363–375. MR 3158562
Paul H. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1980), no. 2, 171–180. MR 583216, DOI 10.1016/0012-365X(80)90033-3
Uwe Franz, Monotone and Boolean convolutions for non-compactly supported probability measures, Indiana Univ. Math. J. 58 (2009), no. 3, 1151–1185. MR 2541362, DOI 10.1512/iumj.2009.58.3578
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, DOI 10.1007/978-3-642-39459-1_{8}
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, DOI 10.7146/math.scand.a-15023
Takahiro Hasebe, Free infinite divisibility for beta distributions and related ones, Electron. J. Probab. 19 (2014), no. 81, 33. MR 3256881, DOI 10.1214/EJP.v19-3448
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, DOI 10.1142/S0219025710003973
Takahiro Hasebe and Alexey Kuznetsov, On free stable distributions, Electron. Commun. Probab. 19 (2014), no. 56, 12. MR 3254735, DOI 10.1214/ECP.v19-3443
Takahiro Hasebe and Noriyoshi Sakuma, Unimodality of Boolean and monotone stable distributions, Demonstr. Math. 48 (2015), no. 3, 424–439. MR 3391380, DOI 10.1515/dema-2015-0030
T. Hasebe and S. Thorbjørnsen, Unimodality of the freely selfdecomposable probability laws, J. Theoret. Probab., to appear, DOI 10.1007/s10959-015-0595-y.
Melanie Hinz and Wojciech Młotkowski, Free powers of the free Poisson measure, Colloq. Math. 123 (2011), no. 2, 285–290. MR 2811179, DOI 10.4064/cm123-2-11
H.-W. Huang, Supports, regularity and $\boxplus$-infinite divisibility for measures of the form $(\mu ^{\boxplus p})^{\uplus q}$, arXiv:1209.5787v1
Wissem Jedidi and Thomas Simon, Further examples of GGC and HCM densities, Bernoulli 19 (2013), no. 5A, 1818–1838. MR 3129035, DOI 10.3150/12-BEJ431
Zbigniew J. Jurek, Relations between the $s$-self-decomposable and self-decomposable measures, Ann. Probab. 13 (1985), no. 2, 592–608. MR 781426
A. Ya. Khintchine, On unimodal distributions, Izv. Nauk Mat. i Mekh. Inst. Tomsk 2 (1938), 1–7 (in Russian).
Hans Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (1992), no. 2, 409–438. MR 1165862, DOI 10.1016/0022-1236(92)90055-N
Naofumi Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 3, 337–371. MR 2016316, DOI 10.1142/S0219025703001365
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, DOI 10.1023/A:1008643104945
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, DOI 10.1017/CBO9780511735127
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
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, DOI 10.1007/s10959-010-0288-5
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, DOI 10.1214/ECP.v12-1274
K. Sato, Lévy Processes and infinitely divisible distributions, Cambridge Studies in Advanced Math. 68, 1999.
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, DOI 10.1090/memo/0627
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
Richard P. Stanley, Parking functions and noncrossing partitions, Electron. J. Combin. 4 (1997), no. 2, Research Paper 20, approx. 14. The Wilf Festschrift (Philadelphia, PA, 1996). MR 1444167
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
Gabriel H. Tucci, Limits laws for geometric means of free random variables, Indiana Univ. Math. J. 59 (2010), no. 1, 1–13. MR 2666470, DOI 10.1512/iumj.2010.59.3775
Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, DOI 10.1016/0022-1236(86)90062-5
Dan Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), no. 2, 223–235. MR 915507
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, DOI 10.1007/BFb0074909
Jiun-Chau Wang, Strict limit types for monotone convolution, J. Funct. Anal. 262 (2012), no. 1, 35–58. MR 2852255, DOI 10.1016/j.jfa.2011.09.004
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, DOI 10.1090/mmono/065