Tree convolution for probability distributions with unbounded support

TL;DR

This paper introduces a complex-analytic framework for tree-based convolutions of probability measures with unbounded support, generalizing various non-commutative convolutions and establishing limit theorems akin to classical stable laws.

Contribution

It defines new tree convolutions for probability measures using fixed-point equations and proves a general limit theorem for their iterated convolutions, extending free probability results.

Findings

Defined tree convolutions via fixed-point equations for Cauchy transforms.

Proved a limit theorem for iterated tree convolutions.

Derived limit laws for measures in the domain of attraction of stable laws.

Abstract

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in "An operad of non-commutative independences defined by trees" (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $\mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $\boxplus_{\mathcal{T}}(\mu_1,\dots,\mu_N)$ for arbitrary probability measures $\mu_1$, ..., $\mu_N$ on $\mathbb{R}$ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from doi:10.4064/dm797-6-2020. We prove a general limit theorem for iterated $\mathcal{T}$-free convolution similar to Bercovici and Pata's results in the free case in "Stable laws and domains of attraction in free probability" (Annals of Mathematics, 1999, doi:10.2307/121080), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.