Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation


In a previous paper (Benaych-Georges in Related Convolution 2006), we defined the rectangular free convolution ⊞λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞λ -infinitely divisible distributions, which preserves limit theorems. We give an interpretation of this correspondence in terms of random matrices: we construct distributions on sets of complex rectangular matrices which give rise to random matrices with singular laws going from the symmetric classical infinitely divisible distributions to their ⊞λ-infinitely divisible correspondents when the dimensions go from one to infinity in a ratio λ.

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


  1. 1.

    Akhiezer, N.I.: The classical moment problem, Moscou (1961)

  2. 2.

    Barndorff-Nielsen O.E. and Thorbjørnsen S. (2002). Self decomposability and Levy processes in free probability. Bernoulli 8(3): 323–366

  3. 3.

    Benaych-Georges, F.: Classical and free infinitely divisible distributions and random matrices. Ann. Probab. 33(3):1134–1170 available on the web page of the author (2005)

  4. 4.

    Benaych-Georges, F.: Rectangular random matrices. Related Convolution (submitted), available on the web page of the author

  5. 5.

    Benaych-Georges, F.: Rectangular random matrices, related free entropy and free Fisher’s information (submitted), available on the web page of the author

  6. 6.

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

  7. 7.

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

  8. 8.

    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

  9. 9.

    Cabanal-Duvillard T. (2005). A matrix representation of the Bercovici–Pata bijection Electron. J.~Probab. 10(18): 632–66

  10. 10.

    Donoghue W. (1974). Monotone matrix functions and analytic continuation. Springer, Berlin Heidelberg New York

  11. 11.

    Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1966)

  12. 12.

    Gnedenko V. and Kolmogorov A.N. (1954). Limit Distributions for Sums of Independent Random Variables. Adisson-Wesley, Cambridge

  13. 13.

    Hiai F, Petz D (2000) The semicircle law, free random variables, and entropy. Amer. Math. Soc., Mathematical Surveys and Monographs, vol. 77

  14. 14.

    Nelson E. (1974). Notes on non-commutative integration. J. Funct. Anal. 15: 103–116

  15. 15.

    Oravecz F. and Petz D. (1997). On the eigenvalue distribution of some symmetric random matricesorpetz [OP97]. Acta Sci. Math. (Szeged) 63(3–4): 383–395

  16. 16.

    Petrov, V.V.: Limit theorems of probability theory Oxford Studies in Probability, 4 (1995)

  17. 17.

    Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer. Math. Soc. 132(627) (1998)

  18. 18.

    Voiculescu, D.V., Dykema, K., Nica, A.: Free Random Variables. CRM Monograghs Series No. 1, Amer. Math. Soc., Providence (1992)

Download references

Author information

Correspondence to Florent Benaych-Georges.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benaych-Georges, F. Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation. Probab. Theory Relat. Fields 139, 143–189 (2007).

Download citation


  • Random matrices
  • Free probability
  • Free convolution
  • Marchenko–Pastur distribution
  • Infinitely divisible distributions

Mathematics Subject Classifications (2000)

  • 15A52
  • 46L54
  • 60E07
  • 60F05