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Contents of PMS, Vol. 40, Fasc. 2,
pages 245 - 267
DOI: 10.37190/0208-4147.40.2.4
Published online 24.6.2020
 

Free infinite divisibility for generalized power distributions with free Poisson term

Junki Morishita
Yuki Ueda

Abstract: We study free infinite divisibility (FID) for a class of generalized power distributions with free Poisson term by using complex analytic methods and free cumulants. In particular, we prove that (i) if \(X\) follows the free generalized inverse Gaussian distribution, then the distribution of \(X^r\) is FID when \(|r|\ge1\); (ii) if \(S\) follows the standard semicircle law and \(u> 2\), then the distribution of \((S+u)^r\) is FID when \(r\le -1\); (iii) if \(B_p\) follows the beta distribution with parameters \(p\) and \(3/2\), then (iii-a) the distribution of \(B_p^r\) is FID when \(|r|\ge 1\) and \(0<p\le 1/2\); (iii-b) the distribution of \(B_p^r\) is FID when \(r\le -1\) and \(p>1/2\).

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