Probability distributions with rational free $R$-transform

TL;DR

This paper characterizes probability distributions with rational free R-transforms, showing their stability under various free convolution operations and providing conditions and examples, including free deconvolution.

Contribution

It introduces the class al M_{ratio} of distributions with rational free R-transforms and establishes their closure properties and conditions for rationality.

Findings

The class al M_{ratio} is closed under free convolution and powers.

A sufficient condition for a rational function to be a free R-transform is provided.

Examples include applications to free deconvolution.

Abstract

We study the class $\mathcal{M}_{\mathrm{ratio}}$ of those probability distributions for which the free $R$-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the monotone convolution. We prove a sufficient condition that a rational function is the free $R$-transform of a probability distribution. Several examples are provided, including that of free deconvolution.