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1

Urbanik type subclasses of free-infinitely~divisible~transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 1

23--39

EN

For the class of free-infinitely divisible transforms we introduce three families of increasing Urbanik type subclasses. They begin with the class of free-normal transforms and end up with the whole class of free- infinitely divisible transforms. Those subclasses are derived from the ones of classical infinitely divisible measures for which random integral repre- sentations are known. Special functions like Hurwitz–Lerch, polygamma and hypergeometric functions appear in kernels of the corresponding integral representations.

2

On background driving distribution functions (BDDF) for some selfdecomposable variables

Jurek Zbigniew J.

Mathematica Applicanda

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2021

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Vol. 49, no. 2

85--109

EN

Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via Levy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used for a simulation of those variables. Among the examples discussed are: gamma variables, hyperbolic characteristic functions, Student t-distributions, stochastic area under planar Brownian motions, inverse Gaussian variable, logistic distributions, non-central chi-square, Bessel densities and Fisher z-distributions. Found representations might be of use in statistical applications.

PL

Wiele klasycznych modeli probabilistycznych opiera sie o zmienne losowe samorozkładalne. Maja one losowe reprezentacje całkowe oparte o procesy Lévy’ego. W tej notatce podano wzory dla ich kierujących (generujących) dystrybuant. Takich reprezentacji można używać do symulacji tych zmiennych. Wśród omawianych przykładów są: rozkłady t-Studenta, pole stochastyczne pod planarnymi ruchami Browna, odwrotny rozkład Gaussa, rozkłady logistyczne, niecentralny rozkład chi-kwadrat, rozkład Bessela i rozkłady statystyk Z-Fishera. Podane reprezentację mogą być przydatne w statystyce.

3

On a Relation Between Classical and Free Infinitely Divisible Transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

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2020

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Vol. 40, Fasc. 2

349--367

EN

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.