Products of free random variables and k-divisible partitions

Details Products of free random variables and k-divisible partitions Products of free random variables and k-divisible partitions

2012-01-27

arxiv.org/abs/1201.5825v2

Mathematics

Operator Algebras

Added remarks on Boolean cumulants; Electronic Communications in Probability, vol 17 (2012), no.11

Scientific paper

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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