On the Lukacs property for free random variables
Volume 228 / 2015
Studia Mathematica 228 (2015), 55-72
MSC: Primary 46L54; Secondary 62E10.
DOI: 10.4064/sm228-1-6
Abstract
The Lukacs property of the free Poisson distribution is studied. We prove that if free $\mathbb X$ and $\mathbb Y$ are free Poisson distributed with suitable parameters, then $\mathbb X+\mathbb Y$ and $(\mathbb X+\mathbb Y)^{-{1}/{2}}\mathbb X(\mathbb X+\mathbb Y)^{-{1}/{2}}$ are free. As an auxiliary result we compute the joint cumulants of $\mathbb X$ and $\mathbb X^{-1}$ for free Poisson distributed $\mathbb X$. We also study the Lukacs property of the free Gamma distribution.
