Geodesic
On the Lukacs property for free random variables
Kamil Szpojankowski1
1Wydział Matematyki i Nauk Informacyjnych Politechnika Warszawska Koszykowa 75 00-662 Warszawa, Poland
Studia Mathematica, Tome 228 (2015) no. 1, pp. 55-72
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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.
DOI : 10.4064/sm228-1-6
Keywords: lukacs property poisson distribution studied prove mathbb mathbb poisson distributed suitable parameters mathbb mathbb mathbb mathbb mathbb mathbb mathbb auxiliary result compute joint cumulants mathbb mathbb poisson distributed mathbb study lukacs property gamma distribution

Kamil Szpojankowski  1

1 Wydział Matematyki i Nauk Informacyjnych Politechnika Warszawska Koszykowa 75 00-662 Warszawa, Poland 
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Kamil Szpojankowski. On the Lukacs property for free random variables. Studia Mathematica, Tome 228 (2015) no. 1, pp. 55-72. doi: 10.4064/sm228-1-6

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