• Bernoulli
  • Volume 19, Number 5B (2013), 2750-2767.

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

Octavio Arizmendi and Takahiro Hasebe

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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 $r\neq2$. Special cases include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if $1\leq r\leq2$.

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Bernoulli, Volume 19, Number 5B (2013), 2750-2767.

First available in Project Euclid: 3 December 2013

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beta distribution free infinite divisibility free Poisson law monotone stable law


Arizmendi, Octavio; Hasebe, Takahiro. On a class of explicit Cauchy–Stieltjes transforms related to monotone stable and free Poisson laws. Bernoulli 19 (2013), no. 5B, 2750--2767. doi:10.3150/12-BEJ473.

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