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V a -Deformed Free Convolution

Author

Listed:
  • Fahad Alsharari

    (Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia)

  • Raouf Fakhfakh

    (Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia)

Abstract

In this article, we study V a -transformation of a measure and of a convolution (denoted by a ) defined for a ∈ R . We provide significant insights into the stability of the free Meixner family of probability measures under V a -transformation. We show that the V a -transformation of measures (of convolutions) of any member of the free Meixner family remains in the free Meixner family. We also present some properties of the Marchenko–Pastur law in connection with a -convolution. In addition, some new limit theorems are proved for the a -convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to V a -deformed free cumulants are presented.

Suggested Citation

  • Fahad Alsharari & Raouf Fakhfakh, 2025. "V a -Deformed Free Convolution," Mathematics, MDPI, vol. 13(4), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:572-:d:1587033
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    References listed on IDEAS

    as
    1. Fakhfakh, Raouf, 2020. "Variance function of boolean additive convolution," Statistics & Probability Letters, Elsevier, vol. 163(C).
    2. Raouf Fakhfakh, 2021. "On some properties of Cauchy-Stieltjes Kernel families," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(4), pages 1186-1200, December.
    3. Michael Anshelevich & Wojciech Młotkowski, 2012. "Semigroups of Distributions with Linear Jacobi Parameters," Journal of Theoretical Probability, Springer, vol. 25(4), pages 1173-1206, December.
    Full references (including those not matched with items on IDEAS)

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