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Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product

Author

Listed:
  • Ayed. R. A. Alanzi

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

  • Shokrya S. Alshqaq

    (Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia)

  • Raouf Fakhfakh

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

Abstract

Let F ( ν j ) = { Q m j ν j , m j ∈ ( m − ν j , m + ν j ) } , j = 1 , 2 , be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν 1 and ν 2 . Introduce the set of measures F = F ( ν 1 ) ⊞ F ( ν 2 ) = { Q m 1 ν 1 ⊞ Q m 2 ν 2 , m 1 ∈ ( m − ν 1 , m + ν 1 ) a n d m 2 ∈ ( m − ν 2 , m + ν 2 ) } . We show that if F remains a CSK family, (i.e., F = F ( μ ) where μ is a non-degenerate compactly supported measure), then the measures μ , ν 1 and ν 2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F ) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F ) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.

Suggested Citation

  • Ayed. R. A. Alanzi & Shokrya S. Alshqaq & Raouf Fakhfakh, 2024. "Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product," Mathematics, MDPI, vol. 12(22), pages 1-9, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3465-:d:1515382
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    References listed on IDEAS

    as
    1. Koudou, A. E. & Pommeret, D., 2002. "A Characterization of Poisson-Gaussian Families by Convolution-Stability," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 120-127, April.
    2. Włodzimierz Bryc & Abdelhamid Hassairi, 2011. "One-Sided Cauchy–Stieltjes Kernel Families," Journal of Theoretical Probability, Springer, vol. 24(2), pages 577-594, June.
    3. Fakhfakh, Raouf, 2020. "Variance function of boolean additive convolution," Statistics & Probability Letters, Elsevier, vol. 163(C).
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