Skip to main content
SpringerLink
Log in

Semi-stable distributions in free probability theory

  • Published:
Science in China Series A Aims and scope Submit manuscript

Abstract

Semi-stable distributions, in classical probability theory, are characterized as limiting distributions of subsequences of normalized partial sums of independent and identically distributed random variables. We establish the noncommutative counterpart of semi-stable distributions. We study the characterization of noncommutative semi-stability through free cumulant transform and develop the free semi-stability and domain of semi-stable attraction in free probability theory.

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

References

  1. Gnedenko, B. V., Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables, Reading: Addison-Wesley Publishing Company, 1968.

    Google Scholar 

  2. Meerschaert, M. M., Scheffler, H. P., Series representation for semistable laws and their domain of semistable attraction, J. Theoret. Probab., 1996, 9: 931–959.

    Article  MathSciNet  Google Scholar 

  3. Bercovici, H., Pata, V., Stable laws and domains of attraction in free probability theory, Annals Mathematics, 1999, 149: 1023–1060.

    MathSciNet  Google Scholar 

  4. Barndorff-Nielsen, O. E., Thorbjørnsen, S., Self-decomposability and Lévy Processes in Free Probability, Bernoulli, 2002, 8(3): 323–366.

    MathSciNet  Google Scholar 

  5. Mejzler, D., On a certain class of infinitely divisible distributions, Israel J. Math., 1973, 16: 1–19.

    MATH  MathSciNet  Google Scholar 

  6. Maejima, M., Semi-stable distributions, in Lévy Processes-Theory and applications (eds. Barndorff-Nielsen, O. E., Mikosch, T., Resnick, S. I.), Boston: Birkhäuser, 2001.

    Google Scholar 

  7. Voiculescu, D. V., Dykema, K. J., Nica, A., Free Random Variables, CRM Monograph Series, Vol. 1, A.M.S., 1992.

  8. Voiculescu, D. V., Addition of certain non-commuting random variables, J. Funct. Anal., 1986, 66: 323–346.

    Article  MATH  MathSciNet  Google Scholar 

  9. Rudin, W., Functional Analysis, 2nd ed., New York: McGraw-Hill Inc., 1991.

    Google Scholar 

  10. Maassen, H., Addition of freely independent random variables, J. Funct. Anal., 1992, 106: 409–438.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bercovici, H., Voiculescu, D. V., Free donvolution of measures with unbounded support, Indiana Univ. Math. J., 1993, 42: 733–773.

    MathSciNet  Google Scholar 

  12. Nica, A., R-transforms of free joint distributions and non-crossing partitions, J. Funct. Anal., 1996, 135: 271–296.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shandong University, Ji’nan, 250100, China

    Fan Zhaozhi

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, HH 3050, St. John’s, A1C 5S7, Canada

    Fan Zhaozhi

Authors
  1. Fan Zhaozhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fan, Z. Semi-stable distributions in free probability theory. SCI CHINA SER A 49, 387–397 (2006). https://doi.org/10.1007/s11425-006-0387-z

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-006-0387-z

Keywords

Search

Navigation