21 research outputs found
On the Lukacs property for free random variables
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