Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A limit theorem for products of free unitary operators
Download PDF
Download PDF
  • Published: 21 August 2007

A limit theorem for products of free unitary operators

  • Vladislav Kargin1 

Probability Theory and Related Fields volume 141, pages 603–623 (2008)Cite this article

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

This paper establishes necessary and sufficient conditions for the sequence of products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Diaconis, P.: Group Representations in Probability and Statistics. Institute of Mathematical Statistics, Lecture Notes—Monograph, Series of IMS, vol. 11 (1988)

  2. Dvoretzky A. and Wolfowitz J. (1951). Sums of random integers reduced modulo m. Duke Math. J. 18: 501–507

    Article  MATH  MathSciNet  Google Scholar 

  3. Garnett J.B. (1981). Bounded Analytic Functions, Pure and Applied Mathematics, vol. 96, 1st edn. Academic, New York

    Google Scholar 

  4. Grenander U. (1963). Probabilities on Algebraic Structures. Wiley, New York

    MATH  Google Scholar 

  5. Haagerup U. (1997). On Voiculescus R- and S-transforms for free non-commuting random variables. In: Voiculescu, D.-V. (eds) Free Probability Theory, Fields Institute Communications vol. 12, pp 127–148. American Mathematical Society, Providence

    Google Scholar 

  6. Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables And Entropy, Mathematical Surveys and Monographs, vol. 77, 1st edn. American Mathematical Society, Providence (2000)

  7. Kawada, Y., Itô, K.: On the probability distribution on a compact group. In: Proceedings of Physical and Mathematical Society of Japan, vol. 22. pp. 977–998

  8. Levy P.: L’addition des variables aleatoires definies sur une circonference. In: Dugue, D. (ed.) Oeuvres de Paul Levy, vol 3 pp. 452–492. Gauthier-Villars, Paris,(1976). 1939

  9. Saloff-Coste L. (2004). Random walks on finite groups. In: Kesten H. (eds) Probability on Discrete Structures, Encyclopaedia of Mathematical Sciences, vol. 110, pp. 263–346

  10. Voiculescu, D.: Symmetries of some reduced free product C *-algebras. In: Lecture Notes in Mathematics, vol. 1132, pp. 556–588. Springer, New York (1983)

  11. Voiculescu D. (1986). Addition of certain non-commuting random variables. J. Funct. Anal. 66: 323–346

    Article  MATH  MathSciNet  Google Scholar 

  12. Voiculescu D. (1987). Multiplication of certain non-commuting random variables. J. Oper. Theor. 18: 223–235

    MATH  MathSciNet  Google Scholar 

  13. Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. CRM Monograph series, No.1, A.M.S. Providence (1992)

  14. Vorobev, N.N.: The addition of independent random variables on finite groups. Matematicheskii Sbornik 34, pp. 89–126 (1954) (in Russian)

  15. Whittaker E.T. and Watson G.N. (1927). A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Vladislav Kargin

Authors
  1. Vladislav Kargin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladislav Kargin.

Additional information

The author would like to express his gratitude to Diana Bloom for her help with editing, and to Professor Raghu Varadhan for useful discussions.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kargin, V. A limit theorem for products of free unitary operators. Probab. Theory Relat. Fields 141, 603–623 (2008). https://doi.org/10.1007/s00440-007-0095-9

Download citation

  • Received: 02 January 2007

  • Revised: 23 May 2007

  • Published: 21 August 2007

  • Issue Date: July 2008

  • DOI: https://doi.org/10.1007/s00440-007-0095-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Free probability
  • Free multiplicative convolution
  • Unitary operators
  • Limit theorem

Mathematics Subject Classification (2000)

  • 46L53
  • 46L54
  • 60F05
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature