open access

https://hdl.handle.net/2115/75993

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

英語

monotone stable law

2013

Bernoulli

19(5B)

2750-2767

We consider a class of probability measures $μ^{α}_{s,r}$ 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 $μ^{α}_{s,2}$ as a free compound Poisson law with Lévy measure a monotone α-stable law. This implies the free infinite divisibility of $μ^{α}_{s,2}$. Moreover, when symmetric or positive, $μ^{α}_{s,2}$ has a representation as the free multiplication of a free Poisson law and a monotone α-stable law. We also investigate the free infinite divisibility of $μ^{α}_{s,r}$ for r≠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≤r≤2.

学術雑誌論文

Version of Record

1350-7265