  Based on recent findings by Bourguin and Peccati, we give a fourth moment
type condition for an element of a free Poisson chaos of arbitrary order to
converge to a free (centered) Poisson distribution. We also show that free
Poisson chaos of order strictly greater than one do not contain any non-zero
free Poisson random variables. We are also able to give a sufficient and
necessary condition for an element of the first free Poisson chaos to have a
free Poisson distribution. Finally, depending on the parity of the considered
free Poisson chaos, we provide a general counterexample to the naive
universality of the semicircular Wigner chaos established by Deya and Nourdin
as well as a transfer principle between the Wigner and the free Poisson chaos.
