k-divisible random variables in free probability

Finally, we concentrate on free additive powers of k-symmetric distributions and prove that μ⊞t is a well defined k-symmetric probability measure, for all t>1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ→μk. We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.