On the Lukacs property for free random variables
Tom 228 / 2015
Studia Mathematica 228 (2015), 55-72
MSC: Primary 46L54; Secondary 62E10.
DOI: 10.4064/sm228-1-6
Streszczenie
The Lukacs property of the free Poisson distribution is studied. We prove that if free 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.