Free Generalized Gamma Convolutions

Abreu, Victor Perez; Sakuma, Noriyoshi

doi:10.1214/ecp.v13-1413

Cited by 21 publications

(14 citation statements)

References 16 publications

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“…We note that there are other nontrivial relations involving these four convolutions. For instance, some free 1 2 -stable distributions are infinitely divisible in the tensor sense [16] (this is also the case in the monotone case, while its explicit calculation is not in the literature). Gaussian distributions are also such examples: they are infinitely divisible both in the tensor and free senses [2].…”

Section: Observation On Cauchy Distributionsmentioning

confidence: 99%

Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of Probability Measures

Hasebe

2011

J Theor Probab

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We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.

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Section: Observation On Cauchy Distributionsmentioning

confidence: 99%

Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of Probability Measures

Hasebe

2011

J Theor Probab

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“…One can consider similar problem in free probability and gets the following result (see [32,33]) for free random variables X, Y random variables X + Y and (X + Y ) −1/2 X (X + Y ) −1/2 are free if and only if X, Y have Marchenko-Pastur (free Poisson) distribution with the same rate. From this example one can see our point-it is not the image under BP bijection of the Gamma distribution (studied in [13,27]), which has the Lukacs independence property in free probability, but in this context the free Poisson distribution plays the role of the classical Gamma distribution.…”

Section: Introductionmentioning

confidence: 99%

On Free Generalized Inverse Gaussian Distributions

Szpojankowski

2018

Complex Anal. Oper. Theory

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We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables X, Y where Y has a free Poisson distribution oneX +Y if and only if X has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.

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“…(iv) Pérez-Abreu and Sakuma [20] introduced another type of free gamma distributions, namely the images of the classical gamma distributions under the Bercovici-Pata bijection. They have free Lévy measure in the form:…”

Section: Examples 12 (I)mentioning

confidence: 99%

“…Möbius inversion (see [19]) and (20) only holds as an asymptotic expansion (see [8]). Recall that a sequence…”

Section: Examples 12 (I)mentioning

confidence: 99%

The Normal Distribution Is Freely Self-decomposable

Sakuma

Thorbjørnsen

2017

International Mathematics Research Notices

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The class of selfdecomposable distributions in free probability theory was introduced by BarndorffNielsen and the third named author. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this paper, we prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey-Wimp-Kerov distribution µc is freely selfdecomposable for any c in [−1, 0].The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform.

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Free Generalized Gamma Convolutions

Cited by 21 publications

References 16 publications

Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of Probability Measures

Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of Probability Measures

On Free Generalized Inverse Gaussian Distributions

The Normal Distribution Is Freely Self-decomposable

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