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2012 Products of free random variables and $k$-divisible non-crossing partitions
Octavio Arizmendi, Carlos Vargas
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Electron. Commun. Probab. 17: 1-13 (2012). DOI: 10.1214/ECP.v17-1773


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.


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Octavio Arizmendi. Carlos Vargas. "Products of free random variables and $k$-divisible non-crossing partitions." Electron. Commun. Probab. 17 1 - 13, 2012.


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

zbMATH: 1256.46036
MathSciNet: MR2892410
Digital Object Identifier: 10.1214/ECP.v17-1773

Primary: 46L54
Secondary: 15A52

Keywords: free multiplicative convolution , Free probability , non-crossing partitions


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