On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson 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 Lévy 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 $r\ne 2$. Special cases include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if $1\le r\le 2$.

Published in at http://dx.doi.org/10.3150/12-BEJ473 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Cited by in corpus (3)

• Free infinite divisibility for beta distributions and related ones
• The Free Tangent Law
• Free Subordination and Belinschi-Nica Semigroup