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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.

On a method of introducing free-infinitely divisible probability measures

Jurek Z. J.

Demonstratio Mathematica

2016

Vol. 49, nr 2

236--251

Random integral mappings (…) give isomorphism between the sub-semigroups of the classical (ID, *) and the free-infinite divisible (...) probability measures. This allows us to introduce new examples of such measures, more precisely their corresponding characteristic functionals.

Cauchy transforms of measures viewed as some functionals of Fourier transforms

Jurek Z. J.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

187--200

The Cauchy transform of a positive measure plays an important role in complex analysis and more recently in so-called free probability. We show here that the Cauchy transform restricted to the imaginary axis can be viewed as the Fourier transform of some corresponding measures. Thus this allows the full use of that classical tool. Furthermore, we relate restricted Cauchy transforms to classical com- pound Poisson measures, exponential mixtures, geometric infinite divisibility and free-infinite divisibility. Finally, we illustrate our approach with some examples.