Electronic Communications in Probability

Products of free random variables and $k$-divisible non-crossing partitions

Octavio Arizmendi and Carlos Vargas

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We derive a formula for the moments and the free cumulants of the multiplication of  $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution $\mu^{\boxtimes k}$, given by Kargin, which show that the support grows at most linearly with $k$. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 11, 13 pp.

Accepted: 25 February 2012
First available in Project Euclid: 7 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras
Secondary: 15A52

Free Probability Free multiplicative convolution Non-crossing partitions

This work is licensed under a Creative Commons Attribution 3.0 License.


Arizmendi, Octavio; Vargas, Carlos. Products of free random variables and $k$-divisible non-crossing partitions. Electron. Commun. Probab. 17 (2012), paper no. 11, 13 pp. doi:10.1214/ECP.v17-1773. https://projecteuclid.org/euclid.ecp/1465263144

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