Skip to main content
SpringerLink
Log in

On Free Regular and Bondesson Convolution Semigroups

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

Abstract

Free regular convolution semigroups describe the distribution of free subordinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Lévy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions, and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result.

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. Arizmendi, O., Hasebe, T., Sakuma, N.: On the law of free subordinators. ALEA Lat. Am. J. Probab. Math. Stat. 10(1), 271–291 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)

    Article  MathSciNet  Google Scholar 

  3. Bertoin, J.: Subordinators: examples and applications. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Volume 1717 of Lecture Notes in Math. Springer, Berlin, pp. 1–91 (1999)

  4. Biane, P.: Processes with free increments. Math. Z. 227, 143–174 (1998)

    Article  MathSciNet  Google Scholar 

  5. Borovkov, K., Burq, Z.: Kendall’s identity for the first crossing time revisited. Electron. Commun. Probab. 6, 91–94 (2001)

    Article  MathSciNet  Google Scholar 

  6. Burridge, J., Kuznetsov, A., Kwaśnicki, M., Kyprianou, A.: New families of subordinators with explicit transition probability semigroup. Stoch. Process. Appl. 124(10), 3480–3495 (2014)

    Article  MathSciNet  Google Scholar 

  7. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. In: Advances in Computational Mathematics, pp. 329–359 (1996)

  8. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  9. Hasebe, T., Kuznetsov, A.: On free stable distributions. Electron. Commun. Probab. 19, 12 (2014)

    Article  MathSciNet  Google Scholar 

  10. Kendall, D.G.: Some problems in the theory of dams. J. R. Stat. Soc. Ser. B (Methodol.) 19(2), 207–233 (1957)

    MathSciNet  MATH  Google Scholar 

  11. Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd edn. Springer, Berlin (2014)

    Book  Google Scholar 

  12. Pérez-Abreu, V., Sakuma, N.: Free infinite divisibility of free multiplicative mixtures of the Wigner distribution. J. Theor. Probab. 25(1), 100–121 (2012)

    Article  MathSciNet  Google Scholar 

  13. Sakuma, N.: On free regular infinitely divisible distributions. RIMS Kokyuroku Bessatsu B27, 115–122 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Schilling, R.L., Song, R., Vondracek, Z.: Bernstein Functions: Theory and Applications. De Gruyter Studies in Mathematics, vol. 37. De Gruyter, Berlin (2010)

    Google Scholar 

  15. Zolotarev, V.M.: One-Dimensional Stable Distributions, Volume 65 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1986)

    Book  Google Scholar 

Download references

Acknowledgements

The author would like to thank an anonymous referee for careful reading of the paper and for suggesting several improvements. The research was supported by the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    A. Kuznetsov

Authors
  1. A. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Kuznetsov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuznetsov, A. On Free Regular and Bondesson Convolution Semigroups. J Theor Probab 33, 1493–1505 (2020). https://doi.org/10.1007/s10959-019-00909-w

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-019-00909-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation