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A property of the free Gaussian distribution

  • Raouf Fakhfakh EMAIL logo und Fatimah Alshahrani
Veröffentlicht/Copyright: 3. August 2024
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Georgian Mathematical Journal
Aus der Zeitschrift Georgian Mathematical Journal Band 32 Heft 1

Abstract

Let 𝒦 + ⁢ ( σ ) = { ℙ ( ϑ , σ ) ⁢ ( d ⁢ ζ ) : ϑ ∈ ( 0 , ϑ + ⁢ ( σ ) ) } be the Cauchy–Stieltjes Kernel (CSK) family generated by a probability measure σ which is non degenerate and has support bounded from above. Consider the concept of V a -transformation of measures introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545] for a ∈ ℝ . We prove that V a ⁢ ( ℙ ( ϑ , σ ) ) ∈ 𝒦 + ⁢ ( σ ) for all a ∈ ℝ ∖ { 0 } if and only if the measure σ is of the free Gaussian (semicircle) type law up to affinity.

Keywords: Variance function; Cauchy–Stieltjes transform; free Gaussian distribution
MSC 2020: 60E10; 46L54

Acknowledgements

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (Project No.PNURSP2024R358), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

References

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Received: 2023-08-23
Revised: 2024-02-25
Accepted: 2024-03-04
Published Online: 2024-08-03
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. A note on higher order Dirac operators in Clifford analysis
  3. Action of higher derivations on semiprime rings
  4. Demicompact linear operator. Essential pseudospectra and perturbation
  5. On the rotations and limit cycles of solutions to the basic system of equations
  6. On the criteria of a measure of non-strict cosingularity in the description of spectral properties of operator matrix
  7. A class of nontrivial simple examples of a non-D-space
  8. A study on Fibo-Pascal sequence spaces and associated matrix transformations and applications of Hausdorff measure of non-compactness
  9. A property of the free Gaussian distribution
  10. The influence of c-subnormality subgroups on the structure of finite groups
  11. Wave propagation on hexagonal lattices
  12. Timelike zero mean curvature surfaces in ℝ1 4
  13. A Mazurkiewicz set containing the graph of a Sierpiński–Zygmund function
  14. On corrected Simpson-type inequalities via local fractional integrals
  15. Sobolev regularity for a class of local fractional new maximal operators
  16. On the singular directions of a holomorphic mapping in P n(ℂ)
  17. On minimal surfaces in ℍ2 × ℝ space
  18. Ulyanov inequalities for the mixed moduli of smoothness in mixed metrics
  19. Dunkl-type Segal–Bargmann transform and its applications to some partial differential equations
  20. On generalized derivations in factor rings
  21. Remarks on generalized derivations in factor rings
https://doi.org/10.1515/gmj-2024-2037
Schlagwörter für diesen Artikel
Variance function; Cauchy–Stieltjes transform; free Gaussian distribution

Artikel in diesem Heft

  1. Frontmatter
  2. A note on higher order Dirac operators in Clifford analysis
  3. Action of higher derivations on semiprime rings
  4. Demicompact linear operator. Essential pseudospectra and perturbation
  5. On the rotations and limit cycles of solutions to the basic system of equations
  6. On the criteria of a measure of non-strict cosingularity in the description of spectral properties of operator matrix
  7. A class of nontrivial simple examples of a non-D-space
  8. A study on Fibo-Pascal sequence spaces and associated matrix transformations and applications of Hausdorff measure of non-compactness
  9. A property of the free Gaussian distribution
  10. The influence of c-subnormality subgroups on the structure of finite groups
  11. Wave propagation on hexagonal lattices
  12. Timelike zero mean curvature surfaces in ℝ1 4
  13. A Mazurkiewicz set containing the graph of a Sierpiński–Zygmund function
  14. On corrected Simpson-type inequalities via local fractional integrals
  15. Sobolev regularity for a class of local fractional new maximal operators
  16. On the singular directions of a holomorphic mapping in P n(ℂ)
  17. On minimal surfaces in ℍ2 × ℝ space
  18. Ulyanov inequalities for the mixed moduli of smoothness in mixed metrics
  19. Dunkl-type Segal–Bargmann transform and its applications to some partial differential equations
  20. On generalized derivations in factor rings
  21. Remarks on generalized derivations in factor rings
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