The Normal Distribution Is Freely Self-decomposable

The class of self-decomposable distributions in free probability theory was introduced by Barndorff–Nielsen and Thorbj⊘rnsen. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this article, we prove that the (classical) normal distributions are freely self-decomposable. More generally it is established that the Askey–Wimp–Kerov distribution $\mu_c$ is freely self-decomposable for any $c$ in $[-1,0]$. The main ingredient in the proof is a general characterization of the freely self-decomposable distributions in terms of the derivative of their free cumulant transform.