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arXiv · 1403.5300

On the Lukacs property for free random variables

Kamil Szpojankowski
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Abstract

The Lukacs property of the free Poisson distribution is studied here. We prove that if free $\X$ and $\Y$ are free Poisson distributed with suitable parameters, then $\X+\Y$ and $\left(\X+\Y\right)^{-\frac{1}{2}}\X\left(\X+\Y\right)^{-\frac{1}{2}}$ are free. As as an auxiliary result we give joint cumulants of $\X$ and $\X^{-1}$ for free Poisson distributed $\X$. We also study the Lukacs property of the free gamma distribution.

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BibTeXRIS

Kamil Szpojankowski. 2014-03-20. On the Lukacs property for free random variables. https://doi.org/10.4064/sm228-1-6

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