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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.

Multiply c-decomposable probability measures on R and their characteristic functions

Rajba T.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

443--456

We obtain the characteristic functions of distributions in Lc,α, i.e.α-timesc-decomposable distributions in the class of infinitely divisible distributions,where 0< α ≤∞, 0< c <1. The characteristic functions of α-times selfdecomposable laws (i.e. α-times c-decomposable for each c ∈ (0,1)) are well known (see [3], [5], [9], [13]).

On multiple decomposability of probability measures on R

Rajba T.

Demonstratio Mathematica

2001

Vol. 34, nr 2

275-294

We define multiple decomposable probability measures on R(see [18]) as a generalization of Loeve's ([6], [7]) c-decomposable laws (c G R). We consider multiply decomposability sets as a generalization of Urbanik's decomposability semigroups D(P) ([21]). We characterize Bunge's nested classes of C-decomposable laws ([1)) using the properties of multiply decomposability sets. We give representations of characteristic functions of laws, whose multiply decomposability sets contain certain sets.