Ona relation between classical and free infinitely divisible transforms

Jurek, Zbigniew J.

doi:10.48550/arxiv.1707.02540

Cited by 2 publications

(3 citation statements)

References 9 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the following, for Voiculescu (Nevanlinna) transforms found in Jurek [16], we compute the triplet of characteristics (a X , b X , F[ρ X ,.]) in their corresponding representations (10) via the inversion formula (32).…”

Section: Some Results On Voiculescu Transforms Related To Hyperbolic ...mentioning

confidence: 99%

See 1 more Smart Citation

Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view

Jedidi¹,

Jurek²,

Nuha³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We establish a new connection between the class of Nevanlinna-Pick functions and the one of the exponents associated to spectrally negative Lévy processes. As a consequence, we compute the characteristics related to some hyperbolic functions and we show a property of temporal complete monotonicity, similar to the one obtained via the Lamperti transformation by Bertoin & Yor (On subordinators, selfsimilar Markov processes and some factorizations of the exponential variable, Elect. Comm. in Probab., vol. 6, pp. 95-106, 2001) for self-similar Markov processes. More precisely, we show the remarkable fact that for a subordinator ξ , the function t ↦ t n E[ξ −p t ] is , depending on the values of the exponents n = 0,1,2, p > −1, or a Bernstein function or a completely monotone function. In particular, ξ is the inverse time subordinator of a spectrally negative Lévy process, if, and only if, for some p ≥ 1, the function t ↦ t E[ξ −p t ] is a Stieltjes transform. Finally, we clarify to which extent Nevanlinna-Pick functions are related to free-probability and to Voiculescu transforms, and we provide an inversion procedure.

show abstract

Section: Some Results On Voiculescu Transforms Related To Hyperbolic ...mentioning

confidence: 99%

“…We emphasize that in our approach to free-probability theory, we only use purely imaginary numbers z = iw, w > 1; cf. (32) and Jurek [16,Corollaries 3,4 and 5] and we recall that the hyperbolic characteristic functions were studied:…”

Section: Introductionmentioning

confidence: 99%

Integral transforms related to Nevanlinna-Pick functions from an analytic, probabilistic and free-probability point of view

Jedidi¹,

Jurek²,

Nuha³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…More explicitly, we do it via a random integral mapping K from Jurek (2007) and series representation of a hyperbolic tangent function. In Jurek (2019) an analogy between different notions of infinite divisibility was studied and there were many explicit examples of Pick functions -free-infinite divisible transforms -related to the hyperbolic and Laplace (double exponential) characteristic functions.…”

mentioning

confidence: 99%