Tree Convolution For Probability Distributions With Unbounded Support

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in Jekel and Liu (2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree T of the N -regular tree (with vertices labeled by alternating strings), we define the convolution (sic)(T) (mu(1) ..., mu(N)) for arbitrary probability measures mu(1),..., mu(N) on R using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from Jekel and Liu (2020). We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata's results in the free case Bercovici and Pata (1999), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.