Products of free random variables and k-divisible partitions

Octavio Arizmendi; Carlos Vargas

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arXiv · 1201.5825

Products of free random variables and k-divisible partitions

Octavio Arizmendi·Carlos Vargas

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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 $μ^{\boxtimes k}$, 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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BibTeX RIS

Octavio Arizmendi, Carlos Vargas. 2012-02-27. Products of free random variables and k-divisible partitions. https://arxiv.org/abs/1201.5825

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