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On the Lukacs property for free random variables

Volume 228 / 2015

Kamil Szpojankowski Studia Mathematica 228 (2015), 55-72 MSC: Primary 46L54; Secondary 62E10. DOI: 10.4064/sm228-1-6


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.


  • Kamil SzpojankowskiWydział Matematyki i Nauk Informacyjnych
    Politechnika Warszawska
    Koszykowa 75
    00-662 Warszawa, Poland

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