UNIVERSITY
OF WROC£AW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
44.1 43.2 43.1 42.2 42.1 41.2 41.1
40.2 40.1 39.2 39.1 38.2 38.1 37.2
37.1 36.2 36.1 35.2 35.1 34.2 34.1
33.2 33.1 32.2 32.1 31.2 31.1 30.2
30.1 29.2 29.1 28.2 28.1 27.2 27.1
26.2 26.1 25.2 25.1 24.2 24.1 23.2
23.1 22.2 22.1 21.2 21.1 20.2 20.1
19.2 19.1 18.2 18.1 17.2 17.1 16.2
16.1 15 14.2 14.1 13.2 13.1 12.2
12.1 11.2 11.1 10.2 10.1 9.2 9.1
8 7.2 7.1 6.2 6.1 5.2 5.1
4.2 4.1 3.2 3.1 2.2 2.1 1.2
1.1
 
 
WROC£AW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 40, Fasc. 2,
pages 245 - 267
DOI: 10.37190/0208-4147.40.2.4
Published online 24.6.2020
 

Free infinite divisibility for generalized power distributions with free Poisson term

Junki Morishita
Yuki Ueda

Abstract: We study free infinite divisibility (FID) for a class of generalized power distributions with free Poisson term by using complex analytic methods and free cumulants. In particular, we prove that (i) if \(X\) follows the free generalized inverse Gaussian distribution, then the distribution of \(X^r\) is FID when \(|r|\ge1\); (ii) if \(S\) follows the standard semicircle law and \(u> 2\), then the distribution of \((S+u)^r\) is FID when \(r\le -1\); (iii) if \(B_p\) follows the beta distribution with parameters \(p\) and \(3/2\), then (iii-a) the distribution of \(B_p^r\) is FID when \(|r|\ge 1\) and \(0<p\le 1/2\); (iii-b) the distribution of \(B_p^r\) is FID when \(r\le -1\) and \(p>1/2\).

References


[1] O. Arizmendi and T. Hasebe, On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws, Bernoulli 19 (2013), no. 5B, 2750-2767.

[2] O. Arizmendi, T. Hasebe and N. Sakuma, On the law of free subordinators, ALEA Latin Amer. J. Probab. Math. Statist. 10 (2013), 271-291.

[3] O. E. Barndorff-Nielsen and S. Thorbjørnsen, Lévy laws in free probability, Proc. Nat. Acad. Sci. USA 99 (2002), 16568-16575.

[4] H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory (with an appendix by P. Biane), Ann. of Math. (2) 149 (1999), 1023-1060.

[5] H. Bercovici and D. V. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), 733-773.

[6] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statist. 76, Springer, New York, 1992.

[7] L. Bondesson, A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables, J. Theoret. Probab. 28 (2015), 1063-1081.

[8] N. Eisenbaum, Another failure in the analogy between Gaussian and semicircle laws, in: Séminaire de Probabilités XLIV, Lecture Notes in Math. 2046, Springer, Heidelberg, 2012, 207-213.

[9] C. Goldie, A class of infinitely divisible random variables, Proc. Cambridge Philos. Soc. 63 (1967), 1141-1143.

[10] T. Hasebe, Free infinite divisibility for Beta distributions and related ones, Electron. J. Probab. 19 (2014), no. 81, 33 pp.

[11] T. Hasebe, Free infinite divisibility for powers of random variables, ALEA Latin Amer. J. Probab. Math. Statist. 13 (2016), 309-336.

[12] T. Hasebe and K. Szpojankowski, On the free generalized inverse Gaussian distributions, Complex Anal. Oper. Theory 13 (2019), 3091-3116.

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

[14]A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser. 335, Cambridge Univ. Press, Cambridge, 2006.

[15] V. Pérez-Abreu and N. Sakuma, Free infinite divisibility of free multiplicative mixtures of the Wigner distribution, J. Theoret. Probab. 25 (2012), 100-121.

[16] F. W. Steutel, Note on the infinite divisibility of exponential mixtures. Ann. Math. Statist. 38 (1967), 1303-1305.

[17] K. Szpojankowski, On the Matsumoto–Yor property in free probability, J. Math. Anal. Appl. 445 (2017), 374-393.

[18] O. Thorin, On the infinite divisibility of the Pareto distribution, Scand. Actuar. J. 1977, 31-40.

[19] O. Thorin, On the infinite divisibility of the lognormal distribution, Scand. Actuar. J. 1977, 121-148.

[20] D. V. Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), 323-346.

Download:    Abstract    Full text