Multiplication of free random variables and the S-transform: the case of vanishing mean

N. Raj Rao (MIT)
Roland Speicher (Queen's University)

Abstract


This note extends Voiculescu's S-transform based analytical machinery for free multiplicative convolution to the case where the mean of the probability measures vanishes. We show that with the right interpretation of the S-transform in the case of vanishing mean, the usual formula makes perfectly good sense.

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Pages: 248-258

Publication Date: August 15, 2007

DOI: 10.1214/ECP.v12-1274

References

  1. Hiai, Fumio; Petz, DÈnes. The semicircle law, free random variables and entropy. Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, (2000). Math. Review 1746976
  2. Nadakuditi, Rajesh Rao Applied Stochastic Eigen-Analysis, PhD Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, (2007).
  3. Nica, Alexandru; Speicher, Roland. A "Fourier transform" for multiplicative functions on non-crossing partitions. J. Algebraic Combin. 6 (1997), no. 2, 141--160. Math. Review 1436532
  4. Nica, Alexandru; Speicher, Roland. Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335, Cambridge University Press (2006) Math. Review 2266879.
  5. Rao, N. Raj; RMTool: A random matrix and free probability calculator in MATLAB (2006). Link to software
  6. Rao, N. Raj; Edelman, Alan: The polynomial method for random matrices(2006). Preprint
  7. Voiculescu, Dan. Multiplication of certain noncommuting random variables. J. Operator Theory 18 (1987), no. 2, 223--235. Math. Review 0915507
  8. Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1. American Mathematical Society, Providence, RI, (1992). Math. Review 1217253
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