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1
Content available remote On a method of introducing free-infinitely divisible probability measures
Jurek Z. J.
Demonstratio Mathematica
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2016
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Vol. 49, nr 2
236--251
EN
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.
2
Content available remote Remarks on restricted Nevanlinna transforms
Jankowski L., Jurek Z. J.
Demonstratio Mathematica
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2012
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Vol. 45, nr 2
297-307
EN
The Nevalinna transform (…) of a positive measure (…) and a constant a, plays an important role in complex analysis and - more recently - in the context of the boolean convolution. We show here that its restriction to the imaginary axis, (…), can be expressed as the Laplace transform of the Fourier transform (a characteristic function) of (…). Consequently, (…) is sufficient for the unique identification of the measure (…) and the constant a. Finally, we identify a relation between the free additive Voiculescu (…) and boolean convolutions.
3
Content available remote On relations between Urbanik and Mehler semigroups
Jurek Z. J.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 2
297--308
EN
It is shown that operator-selfdecomposable measures or, more precisely, their Urbanik decomposability semigroups induce generalized Mehler semigroups of bounded linear operators. Moreover, those semigroups can be represented as random integrals of operator valued functions with respect to stochastic Lévy processes. Our Banach space setting is in contrast with the Hilbert spaces on which so far and most often the generalized Mehler semigroups were studied. Furthermore, we give new proofs of the random integral representation.
4
Content available remote A calculus on Lévy exponents and selfdecomposability on Banach spaces
Jurek Z. J.
Probability and Mathematical Statistics
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2008
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Vol. 28, Fasc. 2
271--280
EN
In infinite-dimensional Banach spaces there is no complete characterization of the Lévy exponents of infinitely divisible probability measures. Here we propose a calculus on Lévy exponents that is derived from some random integrals. As a consequence we prove that each selfdecomposable measure can by factorized as another selfdecomposable measure and its background driving measure that is s-selfdecomposable. This complements a result from the paper of Iksanov, Jurek and Schreiber in the Annals of Probability (2004).
5
Content available remote Kazimierz Urbanik (1930-2005)
Jurek Z.J., Rosiński J., Woyczyński W.A.
Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne
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2006
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[Z] 42
103-121
6
Content available remote Cauchy transforms of measures viewed as some functionals of Fourier transforms
Jurek Z. J.
Probability and Mathematical Statistics
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2006
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Vol. 26, Fasc. 1
187--200
EN
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.
7
Content available remote Selfdecomposable laws associated with hyperbolic functions
Jurek Z. J., Yor M.
Probability and Mathematical Statistics
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2004
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Vol. 24, Fasc. 1
181--190
EN
It is shown that the hyperbolic functions can be associated with self-decomposable distributions (in short: SD probability distributions or Lévy class L of probability laws). Consequently, they admit associated background driving Lévy processes Y (BDLP’s Y). We interpret the distributions of Y (1) via Bessel squared processes, Bessel bridges and local times.
8
Content available remote Selfdecomposability perpetuity laws and stopping times
Jurek Z. J.
Probability and Mathematical Statistics
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1999
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Vol. 19, Fasc. 2
413--419
EN
The decomposability property of Lévy class L of probability distributions, on a Banach space, is extended to a family of stopping times associated with background driving Lévy processes (BDLP). As consequences, this allows us to show that all selfdecomposable measures are perpetuity laws and to get a representation of gamma distribution as an infinite product of independent uniform distributions.
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