Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The $ S$-transform of symmetric probability measures with unbounded supports


Authors: Octavio Arizmendi E. and Victor Pérez-Abreu
Journal: Proc. Amer. Math. Soc. 137 (2009), 3057-3066
MSC (2000): Primary 46L54, 15A52
DOI: https://doi.org/10.1090/S0002-9939-09-09841-4
Published electronically: February 16, 2009
MathSciNet review: 2506464
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Voiculescu $ S$-transform is an analytic tool for computing free multiplicative convolutions of probability measures. It has been studied for probability measures with non-negative support and for probability measures having all moments and zero mean. We extend the $ S$-transform to symmetric probability measures with unbounded support and without moments. As an application, a representation of symmetric free stable measures is derived as a multiplicative convolution of the semicircle measure with a positive free stable measure.


References [Enhancements On Off] (What's this?)

  • 1. O. E. Barndorff-Nielsen and S. Thorbjørnsen, Classical and Free Infinite Divisibility and Lévy Processes. In U. Franz and M. Schürmann (Eds.): Quantum Independent Increment Processes II. Quantum Lévy Processes, Classical Probability and Applications to Physics, Lecture Notes in Math. 1866, pp. 33-159, Springer, 2006. MR 2213448 (2007h:60043)
  • 2. F. Benaych-Georges, On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions, arXiv: 0808.3938v1 [math.PR], 28 Aug 2008.
  • 3. H. Bercovici and V. Pata, with an appendix by P. Biane, Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149 (1999), 1023-1060. MR 1709310 (2000i:46061)
  • 4. H. Bercovici and V. Pata, A free analogue of Hinčin's characterization of infinite divisibility, Proc. Amer. Math. Soc. 128 (2000), 1011-1015. MR 1636930 (2000i:46060)
  • 5. H. Bercovici and D. Voiculescu, Lévy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (1992), 217-248. MR 1151559 (93k:46052)
  • 6. H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), 733-773. MR 1254116 (95c:46109)
  • 7. U. Haagerup, On Voiculescu's $ R$- and $ S$-transforms for free non-commuting random variables. In D. Voiculescu (Ed): Free Probability Theory, Fields Institute Communications 12, pp. 127-148, Amer. Math. Soc., Providence, RI, 1997. MR 1426838 (98c:46137)
  • 8. F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs 77, Amer. Math. Soc., Providence, RI, 2000. MR 1746976 (2001j:46099)
  • 9. A. Nica and R. Speicher, A ``Fourier transform'' for multiplicative functions on non-crossing partitions, J. Algebraic Combin. 6 (1997), 141-160. MR 1436532 (98i:46070)
  • 10. A. Nica and R. Speicher, Commutators of free random variables, Duke Math. J. 92 (1998), 553-592. MR 1620518 (99d:46084)
  • 11. A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series 335, Cambridge University Press, Cambridge, 2006. MR 2266879 (2008k:46198)
  • 12. V. Pata, Lévy type characterization of stable laws for free random variables, Trans. Amer. Math. Soc. 347 (1995), 2457-2472. MR 1311913 (96b:46091)
  • 13. N. Raj Rao and R. Speicher, Multiplication of free random variables and the $ S$-transform: The case of vanishing mean, Elect. Comm. Probab. 12 (2007), 248-258. MR 2335895 (2008f:46082)
  • 14. D. Voiculescu, Dual algebraic structures on operator algebras related to free products, J. Operator Theory 17 (1987), 85-98. MR 873463 (88c:46080)
  • 15. D. Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), 223-235. MR 915507 (89b:46076)
  • 16. D. Voiculescu, K. Dykema and A. Nica, Free Random Variables, CRM Monograph Series 1, Amer. Math. Soc., Providence, RI, 1992. MR 1217253 (94c:46133)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L54, 15A52

Retrieve articles in all journals with MSC (2000): 46L54, 15A52


Additional Information

Octavio Arizmendi E.
Affiliation: School of Mathematics, University of Guanajuato, Valenciana, Guanajuato, GTO 36240, Mexico
Email: octavius@cimat.mx

Victor Pérez-Abreu
Affiliation: Research Center for Mathematics, CIMAT, Apartado Postal 402, Guanajuato, GTO 36000, Mexico
Email: pabreu@cimat.mx

DOI: https://doi.org/10.1090/S0002-9939-09-09841-4
Keywords: Multiplicative convolution, free stable distribution, random matrix
Received by editor(s): September 29, 2008
Received by editor(s) in revised form: November 18, 2008
Published electronically: February 16, 2009
Additional Notes: The first author’s research was supported by SNI-CONACYT Grant A. I. 4337 and the Statistics Laboratory of CIMAT
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society