@article{ECP1773,
author = {Octavio Arizmendi and Carlos Vargas},
title = {Products of free random variables and $k$-divisible non-crossing partitions},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {17},
year = {2012},
keywords = {Free Probability; Free multiplicative convolution; Non-crossing partitions},
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.},
pages = {no. 11, 1-13},
issn = {1083-589X},
doi = {10.1214/ECP.v17-1773},
url = {http://ecp.ejpecp.org/article/view/1773}}