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


 
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1. Title Title of document Products of free random variables and $k$-divisible non-crossing partitions
 
2. Creator Author's name, affiliation, country Octavio Arizmendi; Universität des Saarlandes; Germany
 
2. Creator Author's name, affiliation, country Carlos Vargas; Universität des Saarlandes; Germany
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Free Probability; Free multiplicative convolution; Non-crossing partitions
 
3. Subject Subject classification 46L54; 15A52
 
4. Description Abstract 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) First author supported by DFG - Deutsche Forschungsgemeinschaft, Project SP419/8-1. Second author supported by the Mexican National Council of Science and Technology (CONACYT) ref. 214839/310129.
 
7. Date (YYYY-MM-DD) 2012-02-25
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1773
 
10. Identifier Digital Object Identifier 10.1214/ECP.v17-1773
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 17
 
12. Language English=en en
 
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