Skip to main content
SpringerLink
Log in

Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Let I * and I ⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici–Pata bijection between I * and I ⊞. The class type W of symmetric distributions in I ⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ⊞ and the free counterpart under Λ of the positive distributions in I * is established. It is shown that the class type W does not include all symmetric distributions in I ⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I *. Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

℘:

The set of all probability measures on ℝ.

℘+ :

The set of all probability measures on ℝ+.

℘ s :

The set of all symmetric probability measures on ℝ.

\(\mathcal{C}_{\mu}^{\ast}\) :

The classical cumulant transform of the probability measure μ.

G μ :

The Cauchy transform of the probability measure μ.

F μ :

The reciprocal of the Cauchy transform of the probability measure μ.

\(\mathcal{C}_{\mu}^{\boxplus}\) :

The free cumulant (or R−) transform of the probability measure μ.

I * :

The set of all (classical) infinitely divisible distributions on ℝ.

\(I_{+}^{\ast}\) :

The set of all positive (classical) infinitely divisible distributions on ℝ+.

\(I_{s}^{\ast}\) :

The set of all symmetric (classical) infinitely divisible distributions on ℝ.

I ⊞ :

The set of all free infinitely divisible distributions on ℝ.

\(I_{s}^{\boxplus}\) :

The set of all free symmetric infinitely divisible distributions on ℝ.

\(I_{r+}^{\boxplus}\) :

The set of all free regular infinitely divisible distributions on ℝ+.

ℒ(X):

The law of a real random variable X.

ν μ :

The Lévy measure of μ∈I * or I ⊞.

S μ :

The S-transform of the probability measure μ∈℘+.

w b,a :

The Wigner (or semicircle) distribution with mean b and variance a.

w:

The Wigner (or semicircle) distribution with mean 0 and variance 1.

m c :

The Marchenko–Pastur (or free Poisson) distribution with parameter c>0.

m:

The Marchenko–Pastur (or free Poisson) distribution with parameter 1.

a s :

The symmetric arcsine distribution on (−s,s).

a:

The symmetric arcsine distribution on (−1,1).

\(\mathrm{a}_{s}^{+}\) :

The positive arcsine distribution on (0,s).

a+ :

The positive arcsine distribution on (0,1).

γ b,a :

The Gaussian distribution with mean b and variance a.

γ s :

The Gaussian distribution with mean 0 and variance s.

p c :

The classical Poisson distribution with mean c>0.

\(\gamma_{s}^{(2)}\) :

The gamma distribution with shape parameter 1/2 and scale parameter s.

References

  1. Arizmendi, O., Pérez-Abreu, V.: The S-transform of symmetric probability measures with unbounded supports. Proc. Am. Math. Soc. 137, 3057–3066 (2009)

    Article  MATH  Google Scholar 

  2. Arizmendi, O., Barndorff-Nielsen, O.E., Pérez-Abreu, V.: On free and classical type G distributions. Braz. J. Probab. Stat. 24, 106–127 (2010). Special issue 12th Brazilian School of Probability, Ouro Preto, Brazil, 2008

    Article  MathSciNet  MATH  Google Scholar 

  3. Banica, T., Belinschi, B.T., Capitaine, M., Collins, B.: Free Bessel laws. Can. J. Math. (2010, to appear)

  4. Barndorff-Nielsen, O.E., Maejima, M., Sato, K.: Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–33 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Belinschi, S.T., Nica, A.: On a remarkable semigroup of homomorphisms with respect to free multiplicative convolutions. Indiana Univ. Math. J. 57, 1679–1713 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory. Ann. Math. 149, 1023–1060 (1999). With an appendix by P. Biane

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Bercovici, H., Wang, J.-C.: Limit theorems for free multiplicative convolutions. Trans. Am. Math. Soc. 360, 6089–6102 (2008a)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bercovici, H., Wang, J.-C.: On freely indecomposable measures. Indiana Univ. Math. J. 57, 2601–2610 (2008b)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chistyakov, G., Götze, F.: Limit theorems in free probability theory I. Ann. Probab. 36, 54–90 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs, vol. 77. Am. Math. Soc., Providence (2000)

    MATH  Google Scholar 

  12. Hinz, M., Młotkowski, W.: Multiplicative free square of the free Poisson measure and examples of free symmetrization. Colloq. Math. (2010, to appear)

  13. Kelker, D.: Infinite divisibility and variance mixtures of the normal distribution. Ann. Math. Stat. 42, 802–808 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  14. Młotkowski, W.: Combinatorial relation between free cumulants and Jacobi parameters. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12, 291–306 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Młotkowski, W.: Fuss–Catalan numbers in noncommutative probability. Preprint (2010)

  16. Nica, A., Speicher, R.: A “Fourier transform” for multiplicative functions of non-crossing partitions. J. Algebraic Comb. 6, 141–160 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Notes Series, vol. 335. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  18. Pérez-Abreu, V., Rosiński, J.: Representation of infinitely divisible distributions on cones. J. Theor. Probab. 20, 535–544 (2007)

    Article  MATH  Google Scholar 

  19. Raj Rao, N., Speicher, R.: Multiplication of free random variables and the S-transform: the case of vanishing mean. Electron. Commun. Probab. 12, 248–258 (2007)

    MATH  Google Scholar 

  20. Rosiński, J.: On a class of infinitely divisible processes represented as mixtures of Gaussian processes. In: Cambanis, S., Samorodnitsky, G., Taqqu, M. (eds.) Stable Processes and Related Topics, pp. 27–41. Birkhäuser, Boston (1991)

    Google Scholar 

  21. Sakuma, N.: Characterizations of the class of free self decomposable distributions and its subclasses. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12, 51–65 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  23. Steutel, F.W., Van Harn, K.: Infinite Divisibility of Probability Distributions on the Real Line. Dekker, New York (2003)

    Book  Google Scholar 

  24. Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. CRM Monograph Series, vol. 1. Am. Math. Soc., Providence (1992)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Probability and Statistics, CIMAT, Apdo. Postal 402, Guanajuato Gto., 36000, Mexico

    Victor Pérez-Abreu

  2. Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Yokohama, 223-8522, Japan

    Noriyoshi Sakuma

Authors
  1. Victor Pérez-Abreu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Noriyoshi Sakuma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noriyoshi Sakuma.

Additional information

Part of V. Pérez-Abreu’s work was done while the first author was visiting Keio University during his frequent visits there. He acknowledges the support and hospitality of the Mathematics Department of this university.

Part of N. Sakuma’s work was done while the second author was visiting CIMAT. He sincerely appreciates the support and hospitality of CIMAT. He is supported by the Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pérez-Abreu, V., Sakuma, N. Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution. J Theor Probab 25, 100–121 (2012). https://doi.org/10.1007/s10959-010-0288-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-010-0288-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation