Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous p...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2016 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2016
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147733 |
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| Cite this: | Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples / N. Demni // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 28 назв. — англ. |
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Demni, N. 2019-02-15T18:52:39Z 2019-02-15T18:52:39Z 2016 Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples / N. Demni // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C05; 33C20; 33C45; 44A15; 44A20 DOI:10.3842/SIGMA.2016.035 https://nasplib.isofts.kiev.ua/handle/123456789/147733 For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in [0,1] while another example of Beta distributions in [−1,1] is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html. The author would like to thank H. Cohl and C. Dunkl for their helpful remarks on earlier versions of the paper. He also greatly appreciates the suggestions of the anonymous referees which considerably improved the presentation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples |
| spellingShingle |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples Demni, N. |
| title_short |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples |
| title_full |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples |
| title_fullStr |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples |
| title_full_unstemmed |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples |
| title_sort |
generalized stieltjes transforms of compactly-supported probability distributions: further examples |
| author |
Demni, N. |
| author_facet |
Demni, N. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in [0,1] while another example of Beta distributions in [−1,1] is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147733 |
| citation_txt |
Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples / N. Demni // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 28 назв. — англ. |
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2025-12-07T13:31:43Z |
| last_indexed |
2025-12-07T13:31:43Z |
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1850856495908913152 |