On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws

TL;DR

This paper explores a class of probability measures with explicit Cauchy-Stieltjes transforms, revealing their connections to free Poisson laws, beta distributions, and conditions for free infinite divisibility, including new representations and special cases.

Contribution

It introduces a class of measures with explicit transforms, identifies their relation to free Poisson and monotone stable laws, and characterizes their free infinite divisibility properties.

Findings

Includes symmetric beta and free Poisson laws as special cases

Identifies free infinite divisibility conditions for the measures

Provides new representations as free products involving stable laws

Abstract

We consider a class of probability measures $\mu_{s,r}^{\alpha}$ which have explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $\mu_{s,2}^{\alpha}$ as a free compound Poisson law with L\'{e}vy measure a monotone $\alpha$-stable law. This implies the free infinite divisibility of $\mu_{s,2}^{\alpha}$. Moreover, when symmetric or positive, $\mu_{s,2}^{\alpha}$ has a representation as the free multiplication of a free Poisson law and a monotone $\alpha$-stable law. We also investigate the free infinite divisibility of $\mu_{s,r}^{\alpha}$ for $r\neq2$. Special cases include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if $1\leq r\leq2$.