Tree convolution for probability distributions with unbounded support

Abstract

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in [27], 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 ⊞ T (µ 1 , . . . , µ N ) for arbitrary probability measures µ 1 , . . . , µ 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 [27]. We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata's results in the free case [12], and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.