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On a Relation Between Classical and Free Infinitely Divisible Transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

349--367

We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.

Free Infinite Divisibility for Generalized Power Distributions with Free Poisson Term

Morishita Junki, Ueda Yuki

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

245--267

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 Xr is FID when │r│≥1; (ii) if S follows the standard semicircle law and u > 2, then the distribution of (S + u)r is FID when r≤−1; (iii) if Bp follows the beta distribution with parameters p and 3/2, then (iii-a) the distribution of Brp is FID when │r│≥1 and 0 < p≤1/2; (iii-b) the distribution of Brp is FID when r≤−1 and p >1/2.

On free infinite divisibility for classical Meixner distributions

Bożejko M., Hasebe T.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

363--375

We prove that symmetric Meixner distributions, whose probability densities are proportional to |Γ(t + ix)|2, are freely infinitely divisible for 0 < t ≤ 1/2. The case t = 1/2 corresponds to the law of Lévy’s stochastic area whose probability density is 1/cosh(πx). A logistic distribution, whose probability density is proportional to 1/cosh2(πx), is also freely infinitely divisible.